This indeed could be a problem. There is a famous case C. 70 BCE in which Verres, the governor of Sicily, allegedly crucified a man on the accusation of being a spy, who swore up and down he was a Roman citizen. Did the poor fellow no good, although his story helped Cicero get Verres exiled a bit later.

There's also a problem of politics. A citizen from Comum, a transpadine city that got the citizenship from Julius Caesar, got himself whipped by the consul Marcellus, basically to make a point about Julius Caesar giving out citizenship. Citizens were protected from whipping without trial; Julius Caesar's grants can't protect you, fool!

There were documents to refer to though; you did't need to rely on someone believing you or honoring your claims if they did. The censors were supposed to have lists of citizens that noted their tribe, property class, and century. Censors were not always good at keeping these up to date, and during the 1st century the institution fell apart almost entirely.

During the imperial period (I am not sure about during the Republic), citizens living in provinces who had children who could be citizens by birth would appear before a magistrate and ask their kids be included in an album - a list of citizens. This request could be done at the time of birth, or even years afterward if, for instance because of military service, the father was unable to make the application earlier. Unfortunately as far as I know no album exists, but there are records of court cases dealing with them preserved in papyri.

For those who earned citizenship in the army, there were military diplomas that gave the details of service and rights, and the names of people involved - witnesses, the granting authority, the recipient, unit, date the diploma was conferred. These are small enough to carry around like a passport.

All of these documents are only useful if you're allowed to get access to them and present them, and you're not fed to the beasts out of hand.

I used to have an article on Paul's invocation of the Roman citizenship. I can't find it at the moment and I don't remember it that well... if someone has an idea what this is I'd appreciate it.