There's a bit of a discussion to be had about "Intramural" violence in Medieval and Renaissance Italian cities, which I touched upon in this older answer which might interest you.

One of the points of the answer I linked above aims to bring across is that conflict resolution was a serious and complicated thing in Renaissance Italy. Tensions between families, factions, and social categories could and did devolve into violent confrontations, but significant thought and energy went into making sure this was avoided where possible.

While intramural conflicts could potentially devolve into full-scale military-style confrontations, more commonly they took the form of smaller-scale affairs more aligned with our conception of individual vendettas. Outside of popular revolt or wide-scale intramural conflict, the public use of weapons was often reserved for specific purposes, to settle specific scores, and to achieve specific goals. Violence, when it occurred, was destabilizing and scorned; while it was sometimes unavoidable, it was not encouraged. This is why legitimization of public violence was immeasurably important to Renaissance Italians, and it was predictably easier to garner understanding of a single act of violence caused by specific points of contention than it would be to justify an ongoing feud against a whole group of people (although complicating the matter is the altogether common phenomenon whereby a few people accrued enmity in their role as leaders of a group or category of people). But even when conflicts were seen as legitimate and unavoidable, social pressures were nonetheless strongly in favor of peaceful resolution, and indeed public resolution of legitimate wrongs was a major function of the Communal Governments of Italy. In other words, Medieval Italians did not want tempers to seethe to the point where they had to walk around armed. So in a world where there is no Shakespeare but only dusty historians, before "Ancient grudge break[s] to new mutiny" the neighbors would have dragged old Montague and Capulet in front of Prince Escalus (pretending he is also not fictional) and have them come to terms before publicly signing some sort of agreement (or alternatively but less probably, the conflict would have spread and consumed the whole of the city until one side was destroyed or fled).

What this also means is that unless there were two factions openly warring with each other and setting out for blood (granted we do not know the details of the fictional conflict between Montagues and Capulets, and cannot say if this would be the case) Romeo, Mercutio, and Tybalt's real-life counterparts would probably have not been walking around Verona while conspicuously armed. Indeed, in the play I had always assumed they "fought" with rapiers (but don't remember if this is ever specifically mentioned in the text) which is a very 16th/17th century sidearm, and not what Medieval or Renaissance Italians used to commit murder: murder by clubbing or dagger appears much more often than a theatric sword duel. Further, in keeping with the fact that where possible peers pressured feuding parties into peaceful resolutions, feuds ending in murder were often only those with political or social implications: the lynch mob or death while imprisoned appear more often than a dramatic fight to the death in a public place. In other words, while the feud as described in Romeo and Juliet could sensibly warrant moving about while armed, Medieval and Renaissance Italians did not feud in that way.

While this doesn't mean that absolutely no one was armed in Medieval and Renaissance Italy (it might be advisable, for example, for people to be armed while traveling outside the city, while moving about at night, or why not, during the improbable but not impossible kind of conflict described in Romeo and Juliet) people would not be expected to carry weapons in places where they felt safe, especially (as would be common and expected) if moving about in groups: you do see accounts of people defending themselves from would-be-assassinations with daggers tossed or handed to them by a companion who just so happened to bring one along. This signals not only that these people had felt safe enough to move about unarmed, but also that their companion, not being the target of the would-be-assassination (remember the precarious legitimacy of who can be targeted by violence!) felt safe enough to toss their weapon to a friend in need.