I'm going to speak in broad and general terms here, since you've asked a question that concerns a wide area both geographically, as well as temporally and in terms of existing, previous legal practices (conquered Greek or Hellenistic cities, f.e. would have their own extensive bodies of law, while Germans wouldn't, while in other conquered territories local administration was built up ex nihilo, f.e. in Germany on the right bank of the Rhine). Roman law is also not a monolithic body, so this is only a general overview.
Often, newly conquered territories weren't directly subject to Roman laws. That was, broadly speaking, the situation of most land that was conquered by Rome in Italy and during republican times until the 1st century B.C. Those lands that didn't directly become Roman territory (ager romanus) or were settled by Roman colonies (colonia civium Romanorum) were under their own autonomous laws. The allied cities (civitates foederatae, liberae or stipendiariae) were autonomous in internal affairs, while Rome was in control of external matters.
For the most part, this is also true of the Roman conquest of territories in imperial times, say Gaul, Germany, Britain, the Balkans and Danube region. Roman rule initially was rather indirect, focussed on controlling and integrating the leadership and aristocracy of local groups, generally organized along the lines of gentes (peoples) or civitates, citizenships or groups centered around ethnic lines. So in newly conquered Roman territories, Roman law wasn't too relevant for most people, since the local civitates, client kingdoms, tribes or cities would govern their internal affairs largely after their own fashion. Gradually, they would be more integrated into the Roman polity, or new civitates or coloniae might be founded. Colonies would have Roman citizenship and generally have municipal laws modelled after the example of Rome, but generally cities retained a degree of local legal autonomy.
In addition, one thing to understand about Roman law is that it was not so much codified as it was a cumulative body of jurisdiction. There were the famous twelve tables in the beginning in the early republic, but those covered only narrow areas of law and were not applicable to many of the legal questions that would arise later in Rome's history. The most important sources for Roman law - besides the twelve table law and recognized customs - were edicts and judgements arising out of legal practice, as well as statutes or special laws formulated by the public assemblies, the senate or the emperor to deal with very specific circumstances. One of the most important sources for this was the praetor, a high ranking office whose responsibility was judging cases between citizens. In the beginning of his annual term of office, the praetor would publish an edict in which he stated which types of cases, or actions (actiones) he would grant for judication, which remedies he would offer to settle legal disputes, and which not.
De jure, he didn't create new law (that wasn't in his powers), but in effect he made what the Romans called ius honorarium, by choosing which legal remedies he would grant (or was persuaded to create for unforeseen circumstances), he created a body of legal instruments that could be used and referred to in the future. In the conquered provinces, this role was taken over by the governour similarly to how the praetores spoke law at Rome, by their edicts. For the provinces, these edicts by the governour were one important source of law.
A second important source of law for the provinces was the emperor himself, who might either set law by decree, decide to hear cases brought to him by provincials himself or settle legal disputed brought to him by provincials (often the governour himself, of which many examples survive in the letters between Pliny in his function as provincial gouvernor and the emperor Trajan) either by provocation or by letter (epistula), to which the emperor gave answer (which was calles subscription, since he wrote his answer below the letter, and later rescript). Appeal to the governour or emperor was one often used example to settle disputes in the provinces.
So, how were such things made public? Thankfully, we have quite a few examples of these. Usually, they would be inscribed into large bronze tables (many of these legal acts contained a clause that they were to be in aere incisa, inscribed into bronze), and hung up in a public place. This is the case for many municipal laws that survive, especially from Spain, for example the municipal law of the colonia Iulia Genetiva, modern Urso. This is one of the tables containing chapter 61-69, and you can check a translation of the surviving fragmente here. Others are known f.e. from Irni, the lex Irnitana also being interesting in this context. The lex Irnitana contained a clause that specified up to which amounts in dispute the local magistrates were responsible for judication, and when the case should be brought before the provincial gouvernor, and also (in chapter 85), that the local magistrates were require to put up, in legible writing and in a public place, those edicts of the provincial governour according to which they would judge local cases.
Many of these municipal laws show close parallels to Roman civil law, in the modelling, duties, responsibilites of the magistrates and the political organization itself. Edicts by the senate pertaining to Roman subjects were also published in the same way, the most famous example probably being the senatus consultum de Bacchanalibus, which outlawed practice of the Bacchanalian cult throughout Italy (and thus also in allied towns!) and consequently was hewn into bronze tables put up all over Italy, of which one example survives from Tiriolo.
A famous case of imperial judication is well known from several examples found in the Greek provinces. It seems that the provincial governour, the proconsul, had tried to force a Roman senator living in Asia to take a travelling Roman into his home against his will. Against this, the governour petitioned the emperor, and received a harsh reply: "Sacred letters (from the chancellery of the Emperor, sc. imperial ruling). You seem to us to be ignorant of the senatus consultum. For if you had conferred with (your) experts, you would know that a roman senator is not required to take in a guest against his will." This decision seems to have been (unsurprisingly) popular with local senators, and so it was put up in inscription in several places such as Antiochia Pisidia, Ephesos and Paros [f.e. CIL III 14203, 8), and surviving examples have been found in urban contexts, so probably close to senatorial estates who thus communicated to possible uninvited guests that they were not really required to house them. They were also put up in latin as well as in Greek, to prevent people from putting the language barrier forward as an excuse.
To bring this to a conclusion, the Romans were usually quite anxious about having those laws according to which public life was ordered publicly displayed for anyone to see (to read, well, that is an interesting question if you consider that probably only a fifth of people could read, and less comprehend the legalese). Newly conquered people were usually left to govern their own affairs under the patronage of Rome, and cities could be granted their own legal autonomy that would continue throughout imperial times. Relevant laws and legal decisions would be inscribed, either on bronze or sometimes less durable materials like whitewashed wood (which of course do not survive), for anyone to read who was interested and literate. In cases were local law didn't apply, Roman law could be brought in to settle disputes. But since Roman law was a cumulative body of legal decisions, the possibility remained in most cases to petition the local governour or the emperor to pass judgement, and this was done often, and the results (if advantageous), proudly displayed.
Sources and further reading:
D. Johnston, Roman Law in Context (Cambridge 2004)
D. Knibbe - R. Merkelbach, Allerhöchste Schelte (Zwei Exemplare der Sacrae Litterae aus Ephesos), Zeitschrift für Papyrologie und Epigraphik 31, 1978, 229-232.
E. Metzger, Agree to Disagree: Local Jurisdiction in the lex Irnitana, in: A. Burrows, D. Johnston, R. Zimmermann, eds., Judge and Jurist: Essays in Memory of Lord Rodger of Earlsferry (Oxford 2013) 207–225.
K. Tuori, The Emperor of Law. The Emergence of Roman Imperial Adjudication (Oxford 2016)
