Firstly, let's make it very clear that while the Romans may have had laws against the casual bearing of weapons, they did not have laws against the right to protect one's self and one's property in danger. "We grant to all persons the unrestricted power to defend themselves, so that it is proper to subject anyone, whether a private person or a soldier,who trespasses upon fields at night in search of plunder, or lays by busy roads plotting to assault passers-by, to immediate punishment in accordance with the authority granted to all. Let him suffer the death which he threatened and incur that which he intended (CJ 3.27.1). That, however, isn't the only mention of that. Here's another snippet from the Codex Justinianus, "He who, when placed in peril and in doubt of his life, has killed his aggressor or anyone else, ought not to fear malicious prosecution on account of such an act” (CJ 9.16.2).
But these are all about the right to defend one's self, not necessarily the right to bear arms. I was only able to find 2 references to the right to bear arms. Interestingly, the Codex Justinianus copies the Codex Theodosianus word for word and says "Nulli prorsus nobis insciis adque inconsultis quorumlibet armorum movendorum copia tribuatur / Absolutely no one is granted the ability to wield arms of any description whatsoever without our knowledge and consent" (CT 15.15.1), which is CJ 11.47.1 except taken from an earlier edict.
This comes with a caveat, however. The word "movendorum" can be interpreted as both the "bearing" of arms and the "movement" of arms, meaning that it could have restricted the shipment of military supplies. There's even another piece of legislation which discusses the transportation of military supplies, which the above may have been in reference to, "Whenever requisitions are reasonably necessary for the transportation of arms, your highness shall order a letter to be sent to the most eminent prefect which shall indicate to him the quantity of arms and the place from which they are to be transferred, in order that the prefect immediately gather together the most illustrious governors of the province regarding the requisitions that must be made in proportion to the number of arms that are transferred in accordance with your instruction, so that, following the notice sent by your highness, ships or other requisitions are immediately offered at public expense (CJ 11.10.7).
Then in 440 AD Valentinian III decreed, "By this edict we urge one and all, with confidence in the strength of Rome and with the courage with which one’s own ought to be defended, to use, if the occasion demands it, along with one’s close relatives and friends, whatever arms they can against the enemy, preserving public discipline and the dignity attending their station; and to protect, in faithful concert and with a united front, our provinces and their own fortunes. (Nov.Val. IX). This particular edict, however, would have been a response to foreign invaders at the time, thereby urging anyone to take up arms when necessary.
And of course there were strict laws against slaves taking up arms, for obvious reasons, I think (3 Servile Wars during the Roman Republic stuck in the minds of Romans for many generations afterwards -- part of the reason why Marcus Aurelius' allowing of slaves to join the army after a plague wiped out legions was rather poorly received).
I would say that the Romans had strict laws around the creation and movement of arms that served a military function. However it is unclear whether citizens were outright refused the right to bear arms. And with the knowledge that citizens would have been protected against prosecution in the defense of one's self, I would lean towards the notion that perhaps citizens were not in fact denied the right to bear arms outright (though restrictions around arms and military garments within the pomerium -- the sacred boundary of the city - might have played into effect).
With all of this in mind, I would say that some of these laws -- especially CT 15.15.1, which I mentioned first, and CJ 11.10.7 -- would have been used to guard weapons from falling into the wrong hands by restricting movement/shipments of large amounts of weapons.
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Why would barbarians necessarily want Roman weapons?
The Roman weapons were designed with a specific strategy and military doctrine in mind, the standard weapon of the roman legions consisted in short swords called Gladii designed to strike in a stabbing fashion while taking cover behind the huge rectangular shields that we know.
Most barbarian people generally had a very different military doctrine and outside of the roman formations a gladius was not the most desirable of weapons (it was way too short to be useful in a loose formation doctrine for example). Pila (throwing javelins) could have been useful but the barbarians had all the ressources necessary to make them as well as they were basically shortened spears, spear being one of the most basic weapon humanity designed to kill things or people.
Armor wise the Romans used mainly two types of armors : Hamata (mail) that basically everyone knew how to make (the celts used them a lot) and the Segmentata (plate) which was good protection wise but was a bitch to maintain as it tended to rust and needed a blacksmith to make even the smallest repair, hence why most people (including most Roman legionnaires) preferred the mail which could be repaired on the field by the soldier himself with just some length of metallic wire and could be put on easily like a t-shirt.
Toward the end the Roman Empire even started to include more and more barbarian weapons and tactics to their own, so do not assume necessarily that barbarian tactics and weapons were necessarily of lowest quality. Barbarian was a term the Roman used for anyone not Roman, so it included a lot of civilizations and people, some of which were really advanced.