The famous Italian artist Caravaggio was a criminal who was constantly getting in to street fights in 16th century Rome. He even murdered a man in a dispute over a tennis match. Was this kind of street justice the norm in Rome at the time with arguments between men being settled by a sword fights?
Yes. Medieval and Renaissance Italy had a strong culture of ‘honor’, in which insults, feuds, or even just general unrest could and did lead to violence. Poor enforcement by local authorities (or ‘authorities’) meant that often seeking recourse by means of the law was impractical except for civil disputes and matters the government takes an special interest in.
Think about it like this - there are two powerful families that come to hate each other, and the local government is very weak. When each family comes out to fight the other on the streets, they bring their friends, retainers, etc. So that kind of normalizes that kind of public, nonjudicial violence. From there, it’s a pretty short step to unaffiliated individuals settling their scores in the street - especially since you can’t just look up their address easily, so if you catch 'em in the street, you gotta seize the opportunity!
Also, there were generally no policing forces, just town watches (generally part-time militia, not full-time professionals) that probably weren’t interested in risking getting gutted just to stop two randos from agreeing to kill each other - doubly so if they were ‘someone’ locally, as these fights often occurred between, and you had a serious risk of drawing the ire of all their connections in turn by interfering with their business. Generally they weren’t even interested in enforcing the law, just a kind of bare minimum of enforcement presence to stop people who AREN’T consenting to kill each other in the street. Just something to discourage burglars, thieves, and murderers of uninvolved, unarmed people.