It would almost certainly depend upon the city, its environment, and the training of the people.
Early-modern Nuremberg and other German city states maintained a fairly large arsenel of crossbows in order to equip their citizen levies in time of war. This implies that they expected at least part of the citizenry to need equipment. On the other hand, my documentation on this is well into the gunpowder age and I get the impression that they were never really willing to cover the cost of upgrading those to guns. (Though such cities frequently bought canons for use in the defense of their cities). This may imply that there was no longer as serious a need for personal weapons since citizens would most likely bring their own. On the other hand, it could just mean that such cities got more bang for their buck with artillery. (FWIW, the peasant armies in the revolution of 1525 had similar equipment to the nobles' armies, except for their lack of canons. What canons they had came from the towns who supported them. This may be further evidence that non-artillery equipment was widely available in the early modern era.)
On the other hand, the ancient Greeks, Israelites (even the post-exile people of Jerusalem), etc appear to have kept their own weapons. IIRC, it was customary for greek soldiers to provide their own shield, spear, and breastplate when reporting for duty and the Israelites made a habit of beating their plowshares into swords and spears when they were invaded. Those who rebuilt Jerusalem under Nehemiah appear to have had enough swords for every workman to carry one. Similarly, it appears to have been common practice for colonial American militia members to provide their own small arms. The official militia would provide any canons, however. The development of weapons like the bill, etc from farming implements indicates that at least a good amount of citizen conscripts and militias in Europe were required to provide their own equipment.
I think it's more likely that, in typical D&D-land, a militia would provide their own melee weapons but that the militia commander would stockpile ammunition and expensive weapons that would be used in time of war but not ordinarily in the armory. So, an armory would most likely contain expensive weapons like crossbows (35gp for a light crossbow) and maybe some heavier armors (50gp for scale mail, 25 gp for studded leather) and tower shields (50gp each), but guards would bring their own clubs (free), and longspears (5gp), spears (2gp), or morning stars (8gp). The armory would also have LOTS of bolts for the crossbows since their cost adds up quickly and most people would probably not have enough to last long in a siege in their personal supply.
In frontier areas or wealthier areas where private ownership of crossbows, etc could be assumed, the armory might be stocked almost entirely with ammunition.
Ogrork the Mighty said:
Who said there is no guard? Not the original poster (someone skipped English class!). Let's say there wasn't but instead there was a militia. The militia aren't taking the weapons home silly! If they were, they'd need a HECK of a lot of weapons (basically one per person). No, instead the weapons would be stored in an armoury/garrison from which the active militia could draw their weapons. Out in the country or in a small community, militia might maintain their own weaponry. But not in a city; there's simply too many people.