I've been researching about Venetian warfare and came across this website which inspired me to ask the question. According to this website (of perhaps questionable repute), the following is stated:
In 1303, crossbow practice became compulsory for all citizens in the City of Venice.
While compulsory weapons training is one thing, it is another thing to actually be able to own and operate your own weapons. As such, did the Republic of Venice (between 1300 and say 1500 at the latest since the sources I cited are mostly talking about the 1300s) permit its citizens to own, carry and operate arms or was it forbidden?
According to this Guido Ruggiero, Law and Punishment in Early Renaissance Venice, 69 J. Crim. L. & Criminology 243 (1978) however, it states:
"[C]um multe fraudes committantur in civitate ista per nonullas accipientes duas uxores et hoc propter parva pena ...." A.S.V., M.C., Novella, f.80v (1359) and registered with the Avogadori: Adv., MC., Reg. 24/7, f.45r (1359). The council responsible for overseeing this provision was the Signori di Notte. One of the primary policing agencies of the city, it also enjoyed the right of imposing summary justice in the streets for minor brawls and illegally carrying weapons.
The bold has been inserted by me. This implies that there is a legal way to carry them. That source also states:
A good example of this complexity is revealed by an adjustment of the requirement for a gratia concerned with carrying or using of weapons:
Nor may a gratia be given to anyone who incurred any penalty from the Signori di Notte or the Capi di Sestiere or the Cinque alle Pace [all policing bodies with the right to give summary justice for minor brawling and carrying weapons without the vote of 30 members of the Council of Forty ... nor without the approval of the customary number of Ducal councilors (five of six) ... nor without the approval of five of six of the Capi di Sestiere when the - penalty was imposed by them.
So based on this, it seems that it was possible albeit with a really difficult process, but the question still remains as its not clear whether this actually applies to a common person or not.