I'll have a stab!
Judea in the first century was a Roman province, and as such fell within the ambit of the Roman law. Thus, any action would have been made under the Roman legal system, which differentiated heavily between citizens and non-citizens. The first question to ask is, were both of the parties (the pigheard and the mystic) Roman citizens? If yes, then the claim would fall under the ius civile, or the Civil Law. The ius civile, which had pre-Roman Republic origins, had become highly developed by the 1st century, with the principal sources of law being statutes [leges; singular lex] made by the legislative assemblies, and the Edict made by the Praetors (sort of like an Attorney-General, who held office for a year), which detailed the formula for various actions. There were two Praetors, who each had a different Edict: the Urban Praetor was in charge of Roman citizens, and the Peregrine Praetor was in charge of peregrines (non-citizens).
One key thing about Roman law is that it is very much based around actions [actiones]: you had to go before the Praetor (who would often appoint an iudex, a judge) with a specific action for what you were claiming, which involved speaking specific words known as the formula, which would be unique to each action. These actions must be taken from a closed list of actions stated in the Praetor's Edict, which until about 135AD (during Hadrian's reign) was revised every year. Some examples include the actio emptii (for a buyer to sue a fraudulent/non-performing seller under the law of sale), the actio mandati (to claim performance for a mandatum - a particular type of contractual promise) , and the vindicatio (for an owner of a property to claim his property back from someone who has taken it against his will).
The action you would press for someone drowning your pigs would be for the actio Legis Aquiliae: an action based on the Lex Aquilia, a statute promulgated in the 3rd century BCE, that awarded compensation for specific acts that caused damage to property.
Gaius, a 2nd-century Jurist, explains in his Institutes (a textbook of law):
The action for unjust damage, (damni iniuriae) is established by means of the Aquilian law (lex Aquilia), in the first chapter of which it is provided, that if anyone has illegally killed another's slave, or one of those quadrupeds that are accounted as cattle (pecudes) he shall be condemned to pay to the owner that amount which was the highest value of the thing at any time during the previous year.
(Gaius, Institutes, 3.210-4)
On the question of whether pigs were included in the category of pecudes, Justinian's Institutes (essentially a revised version of Gaius' Institutes with many portions copied verbatim, compiled in 535 AD) has this to say:
Swine are also held to be included, since they come under the term cattle, for they feed in flocks, as Aelius Marcianus notes in his Institutes that Homer, in the Odyssey, [refers to pigs as cattle].
(Justinian, Institutes, 4.3.1)
Therefore, you would go before the praetor pleading an actio legis Aquiliae, and if your evidence was accepted before the praetor (or more likely, an iudex or representative), the mystic would be compelled to pay you the highest value of your pigs in the previous year.
Here’s the really interesting part: it’s possible, in theory, to make a profit through such an action. For instance, if your pigs became diseased and wasted in the past 2 weeks (such as from being possessed by demons), you could still claim for their highest value in the past year, which is likely more than what they would’ve fetched you if you sold them on the open market. This might seem a strange concept that is offensive to principles of equity in many modern legal systems, but to the Romans this made perfect sense.
This is because the Lex Aquilia covered what were known as delicts: something that was a combination of tort and crime, in that it was meant to be both punitive and compensatory. It didn’t matter that you could make a profit from the mystic’s actions – the law was designed to punish the mystic as much as it was designed to compensate you, especially since the Roman criminal legal system at the time wasn’t very developed.
Note that all of the above governs only Roman citizens, who were under the ambit of the Urban Praetor and the ius civile. Non-citizens (peregrines) were under the ambit of the Peregrine Praetor, whose Edict has mostly been lost to history. We do know, however, that some of the peregrine actions were similar or analogous to the ius civile actions. Professor David Daube (a leading scholar of Roman Law, and former Regius Professor of Civil Law at Oxford) has this to say (citation given at the end of this post):
it may be observed that certain features of the law affecting peregrines do look as if they had originated in the court of the urban praetor. For example, in a case of theft or damage to property, a peregrine sues or is sued by an action feigning him to be a citizen...
We can thus assume that even if both parties were peregrines, the action and remedies would've been similar.
EDIT: Some questions below have brought up interesting points, which I'll paste here.
On Intent
The Lex Aquilia did not have provisions for intent, and was strict liability. This is in contrast with theft (furtum) and insult/outrage (iniuria), which did require theftuous/contumelious intent. A likely explanation is that Lex Aquilia liabilities are perhaps meant to be more compensatory than punitive, as evidenced by its damage for the value of the thing damaged or killed, as opposed to theft, which compensates for anything between twofold to fourfold the value of the thing stolen.
As such, it didn't matter if the mystic intended to kill the pigs - the only relevant question was whether he did kill the pigs; whether he caused the pigs to die. If he did, then he was liable under the Lex Aquilia.
On Direct/Indirect Actions
A very interesting point to note is that the Roman law did distinguish between direct and indirect actions. Gaius tells us in his Institutes that the actio legis Aquiliae only lay if "a person caused the damage by his own bodily act", such as stabbing the pig to death. If instead the damage "was caused in any other way," such as if a person has "shut up another's cattle, so that they perished by starvation," this was considered an indirect action. In this case, the actio legis Aquiliae itself would not be available, but an equitable action [actio utilis] would instead be given. [Gaius, Institutes, 3.219]
The difference is one of semantics in pleadings - the remedy was the same, and indeed the difference between direct and indirect actions seems very arbitrary. For instance, someone who throws another's slave into a river, causing the slave to drown, is considered to have caused the death directly rather than indirectly. Gaius states that this is "not difficult to understand", but to me it seems arbitrary.
On one view, perhaps the correct action would be the actio utilis, rather than the actio legis Aquiliae.
Sources:
Gaius, Institutes (tr. T Lambert Mears, 1882, London: Stevens and Sons)
Justinian, Institutes (tr. T Lambert Mears, 1882, London: Stevens and Sons)
B. Nicholas, An Introduction to Roman Law (1962, Oxford: Clarendon Press)
W. Buckland, A Textbook of Roman Law from Augustus to Justinian (3rd ed., 1975, Cambridge: Cambridge University Press)
P. Birks, The Roman Law of Obligations (2014, Oxford: Oxford University Press)
D Daube, The Peregrine Praetor, The Journal of Roman Studies, Vol. 41, Parts 1 and 2 (1951), pp. 66-70
This is a bit of a stretch but I'll do the best I can. Although I am not familiar with any written or case law that has survived from the first century, I am familiar with Roman civil law that was published in the 6th century AD. Although the passage of time between first century Roman Judea and 6th century Roman Judea may have changed the relevant Roman laws, the Corpus Juris Civilis (CJC) of the 6th century was based on a long history of Roman legislation and can hopefully give us some insight into how the Roman legal system handled civil cases like this.
To answer your first question: yes, you could press charges. The exact process is not clear but cases could be brought to a variety of Imperial officials with the Praetorian Prefect being the highest judicial official outside of the Emperor himself by the 6th century. This excerpt from book 1 of CJC presents officials that had judicial power, and how they were expected to make use of this responsibility:
No illustrious Governor of a province, consular ruler, magistrate, or anyone who has received the insignia of the administration of a higher office, that is to say, the illustrious proconsuls, augustal prefects, Counts of the East, or deputies of provinces, or any general or commander of a division of the army, or any count of the Imperial Palace, shall after he has been succeeded, presume to leave the place which he is known to have ruled before the prescribed term of fifty days has elapsed; but, during that time, Governors and consular rulers, the distinguished judges civil as well as military, and other magistrates, shall reside openly in the principal city where they have exercised jurisdiction, and shall not conceal themselves in their own houses, or on the frontiers, or in any fortified buildings, but shall appear before all the persons whom they formerly governed, and in the most public places; so that every one may have ample opportunity to file complaints against them, for theft or other crimes; and that everyone may be defended from injury by his successor; and if he should be accused, it will be at the risk of his office, as well as of that of the curia and the defender of the city; and, having given his oath as security, after he has been brought into court, he can defend himself against those who complain of him (as already stated), and protect himself by means of the law.
In other words, Governors, Consuls, magistrates, deputies, palace officials or even military commanders were not only able to hear cases and give judgements but were punished for shirking this duty. They were expected to make themselves accessible to the public and give everyone a chance to file complaints including civil ones like theft.
Assuming you were able to bring your case against one of these local officials the most closely related guidance I could find was robbery:
Robbery is chargeable also as theft; for who deals with the property of another more against that other's will than the robber? And thus the description of the robber as an audacious thief is a good one. However, as a special remedy for this offence the praetor has introduced the action for robbery, or rapine with violence, which may be brought within a year for four times the value, after a year for simple damages, and while lies even when only a single thing of the slightest value has been taken with violence. This fourfold value, however, is not all penalty, nor is there an independent action for the recovery of the property or its value, as we observed was the case in the action of theft detected in the commission; but the thing or its value is included in the fourfold, so that, in point of fact, the penalty is three times the value of the property
Based on this, it looks like bringing the case against a judge would lead to awarded damages of roughly 4 times the value of the drowned pigs. However, since this case has multiple pigs involved which may have some greater value than just their individual summed value, the CJC suggests taking this into account:
one must not only take account, in the way we have described, of the value of the body of the slave or animal killed, but must also consider all other loss which indirectly falls upon the plaintiff through the killing. For instance, if your slave has been instituted somebody's heir, and, before he has by your order accepted, he is slain, the value of the inheritance you have missed must be taken into consideration; and so, too, if one of a pair of mules, or one of four chariot horses, or one of a company of slave players is killed, account is to be taken not only of what is killed, but also of the extent to which the others have been depreciated.
Unfortunately it is rather unclear what would happen if the mystic wasn't able to afford the legally awarded damages.
Source: The Institutes of Justinian Author: Caesar Flavius Justinian Translator: J.B. Moyle
edit: credit to u/Cowtheduck for a more period relevant answer.
Another edit, the translation I was using isn't great, after looking at the latin the punishment would likely come down as follows: if the pigs were taking by force, vi bonorum raptorum, a fourfold penalty; if the pigs were taken non-violently and the mystic was caught in the act, manifesti furti, a fourfold penalty; or if the pigs were taken non-violently and the mystic was not caught, nec manifesti furti, twofold penalty.
While the question is amusing and many of the answers are interesting, it seems the question is based on the story of the exorcism of the Gerasene (or Gadarene) demoniac, found in Mark 5:1-20, with cognates in Matthew 8:28-34 and Luke 8:26-39 . The problem here is that the question asks about the legal recourse of "a first century Judaean pig farmer", but the story it is based on is quite clearly set not in Judea, but in either Galilee or the Decapolis. Mark 4:35-41 has Jesus calming a storm on the Sea of Galilee and then the exorcism story that follows directly from this says he "came to the other side of the sea, to the country of the Gerasenes" (Mark 5:1). The Matthean version follows the same sequence, with Jesus calming the storm on the Sea of Galilee (Matt 8:23-27) and then performing the exorcism (Matt 8: 28-34), though in that version there are two possessed men, not one.
The Marcan and Matthean versions also differ on which town the demoniacs come from - it's " the country of the Gerasenes" (i.e. Gerasa) in gMark and " the country of the Gadarenes" (i.e. Gadara) in gMatt. This further complicated by the fact that neither Gerasa or Gadara were on the shores of the Sea of Galilee - the former was 10 kms to the south east of the lake and the latter about 50 kms in the same direction.
That aside, there is no doubt that the story is not set anywhere near Judea, which is much further south. If the writers were simply confused about the geography and intended the story to be set on the shore of the lake, then the story is set in Galilee. Galilee was not governed under Roman law, since it was, at the time the story was set, a client kingdom ruled by Herod Antipas. But if the story was actually set in either Gadara or Gerasa, then both of these were part of the group of independent self-governing city states called "the Decapolis", and so would have fallen under the laws of their respective cities. The people of the cities of the Decapolis were a mix of Greek-speaking Syrians, some Romans and a few Jews, but were mainly Semitic-speaking Nabataeans and Arameans. The majority did not practice Judaism and had their own local cults, so the idea they were farming (and eating) pigs, as depicted in the story, makes some sense. In the Marcan version the healed demoniac asks to join Jesus' followers, but Jesus tells him to go home, implying that he was not a Jew. That version then says "he went away and began to proclaim in the Decapolis how much Jesus had done for him; and everyone was amazed" (Mark 5:20)
This means all the answers here based on the idea that this pig farmer was "Judean" and therefore would have recourse to Roman law given Judea was occupied by the Romans are all very worthy, but the story the question is based on is not set in Judea. There is some speculation that the story is something of a subtle joke, since after 70 AD the region was occupied by Legio X Fretensis; a Roman legion whose symbol was a boar or pig. The 2,000 demons driven out in to the pigs give their name as "Legion, for there are many of us". So this story of Jesus driving a "legion" of demons into the sea could be a subtle dig at the occupying Romans and have more to do with the time the gospel of Mark was written (post 70 AD) than anything in Jesus' time, since the region was not under occupation by Roman legionaries earlier in the century.