Virtually all the evidence for this questions is from the Imperial period, after the Roman Army had become regularized. That is -- the first century CE and after. We have good sources for how Imperial soldiers were paid and equipped. The basic answer is that the Roman State provided new recruits with all their basic needs, including clothing and food, but these items were deducted from their pay. There are many itemized documents indicating these deductions (see #1 below). In none of these are weapons or armor specifically mentioned. Nevertheless, the evidence seems to suggest that the average soldier was provided his weaponry by the Roman State upon recruitment and the cost of these items were deducted from his pay, including weapons. Tacitus mentions as much (see #2). It is unclear if that meant they "owned" the state-provided weapons when they were discharged, and no source I can find mentions a soldier going home with his lorica or pilum. We know that many permanent Roman forts along the frontiers had workshops (fabricae) to produce or mend weapons and tack. These were usually small and seemed to be intended for maintenance, but one at Corbridge was very large and may have been a weapons factory for State use. Otherwise, the production of weaponry was left to the private sector, but with the State owning a virtual monopoly on the use of heavy weapons, it must have largely controlled the price. I think we should imagine larger orders, and usually only when disasters happened or new units were being created. It is difficult to imagine how arms production would have worked, if discharged soldiers were carrying off their equipment. Perhaps they "sold" it back to the legion when they left, but I can find no evidence for anything like this. The soldier was also free, however, to supplement or replace these basic weapons and clothing with his own, if he wanted. Pliny the Younger tells us in a letter (Ep. 6.25.3-4) that he gave a centurion named Metilius Crispus 40,000 sesterces for the purchase of equipment. He must have looked very fancy on his way to war, because he never arrived, probably ambushed by robbers.
The basic 'uniform' (habitus) of the Roman soldier in all periods was the sword belt (with pugio dagger), and we hear of severe punishment for soldiers who went discinctus 'unbelted' in the field; going unbelted was often associated with a soft life, or effeminacy (Tacitus Ann. 13.35; Fronto ad verum Imp. 2.1.19; cf the encounter in Petronius, Satyricon 82). This might suggest a more personal connection between a soldier and his sword/dagger, at the very least. But never do we hear about "named" swords, or soldiers wielding antique weapons. The Roman gladius was a durable object, and if properly maintained would have lasted for centuries. In the Republican period, there had to have been legacy weapons in the aristocratic families, yet I can't recall any evidence or mention of family swords, antiques, or anything like this. -- It suggests to me that Roman culture did not fetishize weapons as later cultures would -- there is no Roman Excalibur, no Joyeuse, no Balmung, no Durandal, no Hrunting. If a soldier carried home his sword after he was discharged, I don't imagine it was from any romanticized attachment, even if he carried it for decades. He may have thought of it as just a tool, like his shovel or his spoon. If he did bring it, it might have been more about maintaining his identity as a military man. Sara Elise Phang has an interesting chapter on this subject (Roman Military Service: Ideologies of Discipline in the Late Republic and Early Principate 2008, chapter 3).
1) From a papyrus from Masada (Israel), 72-75 CE (Cotton and Geiger 1989:722)
In the fourth consulship of Emperor [Ves]pas[ian] Augustus (?) Pay Account (?) Gaius Messius, son of Gaius, of the Fabia tribe, from Berytus
I have received as my salary 50 denarii out of this I have paid:
barley [ _ _ _ ]
[ _ _ _ ]rnius
food. . . 20 denarii
boots. . . 5 denarii
leather straps (?). . . 2 denarii
linen tunic. . . 7 denarii
I have received as my salary 60 denarii
out of this I have paid:
barley ? . . . denarii
food . . . 20 denarii
Gaius Antonius
heavy cloak . . . ? denarii
Puplius Valerius
white tunic . . ? denarii
2) The historian Tacitus reports (annals 1.17) that after the death of Augustus, some soldiers in Pannonia put on a mutiny in protest of their "working conditions":
He (Percennius, leader of the mutineers in Pannonia) said that enough harm had been caused over the years by their passivity; old men, several of them with their bodies disfigured by wounds, were serving their thirtieth or fortieth year. There was no end to military service even when they were discharged, since they remained under the standards and performed the same toil under another name. And, if anyone survived all these perils, he was dragged off to a distant country where he received, in the name of a plot of colony land, some waterlogged swamps or uncultivated mountainsides. Indeed military service was relentless and unprofitable; body and spirit were valued at two and a half sesterces a day, and out of this they had to pay for their clothing, weapons, and tents, and bribe vicious centurions to escape routine drudgery.