There's two parts to this question, I think - how do we know what the Romans called each other, and how do we account for the differences between those names and the conventional names of certain famous Romans in modern languages. Once we understand the first, the second becomes much clearer.

Part 1: What Did the Romans Call Each Other?

Knowing what Roman names were is pretty easy - they wrote them down. 'Authentic' as a concept doesn't really work for pre-modern languages - people spelled things as they sounded, which meant that the spelling of even everyday words was flexible from place to place and person to person, and there was no real standard of 'right spelling' versus 'wrong spelling'. However, names tend to be pretty consistent, and there's no real contest over how (say) Julius Caesar would have written out his full name. Unfortunately, that's not quite the end of the story.

You'll often hear that Romans had three names - a praenomen, a nomen and a cognomen, with the nomen corresponding to a modern 'surname' and the cognomen a sort of nickname. This isn't bad as a rule of thumb, but massively oversimplifies the matter. A very good article to get started with is Bennet Salway's 1994 'What's in a Name? A Survey of Roman Onomastic Practice from c. 700 BC to AD 700' in the Journal of Roman Studies.

Essentially - Salway notices that Roman naming practices never really stand still; they evolved constantly over time. The Romans themselves conjectured that their earliest ancestors had only one name, noting that this holds for characters in their earliest myths like Romulus, Numitor and Faustulus, but it seems that by at least the seventh century the Romans, as well as most other Italian peoples, had a binomial system whereby a single personal name was added to a fixed, hereditary family name. Salway makes the point that this is very odd - very few other Indo-European peoples did it, and even the Greeks generally stuck to a single personal name.^(1) These names are generally known as the praenomen and the gentilicum (lit: 'the family one', from gens, 'family').

Perhaps not surprisingly, if you know how obsessed the Romans were with family and heredity, they put the most importance on the gentilicum. Societies using single names tend to make them both meaningful and potentially compoundable - so you end up with Greek names like 'Anaxagoras' ('leader of the marketplace'), 'Protagoras' ('first in the marketplace'), and so on. You also, necessarily, end up with a lot of them - the Lexicon of Greek Personal Names records 35,000 names shared among 300,000 known individuals. Latin praenomina aren't like that: we only know of 65 unique praenomina, and the rather obscure (probably) second-century AD writer Probus, who wrote a book on them, reckoned that only about thirty existed. Even Romans who dug into the subject couldn't find most of those names, and with good reason - 99% of Romans shared 17, usually short, simple and practically meaningless praenomina. This was predictable enough that they could be abbreviated without a problem -so T. Quiscumque is always a Titus, M. Quiscumque is always a Marcus, and so on.^(2)

What this means is that praenomina are not very useful things if you want to tell people apart (as what Salway calls a 'diacritic name' - the name you call yourself, or someone else, that distinguishes them from anyone else) - especially as certain families tended to favour certain praenomina, or to consistently name the first son after his father, which really didn't help. Indeed, the praenomen only really worked within the household, and so it was a mark of (over-)familiarity to call someone simply by that. However, the problem with the gentilicum was twofold. Firstly, there was nothing to stop unrelated families from latching onto the prestige of a well-known gens and using the same gentilicum. Secondly, the gentilica native to Latium developed from patronymics - so all the sons of Marcus become e.g. Gaius Marcius, Lucius Marcius, Titus Marcius, and so on. You can do the maths - with such a restricted pool of personal names, that means a lot of 'Marcii' families going about with a completely legitimate claim to the name and no connection to each other whatsoever.

In friendly contexts, this wasn't a problem - Catullus often calls his friends by their nomina, such as 'Furius' and 'Aurelius', which would be shared by hundreds if not thousands of people in Rome, but only one person in his immediate circle of friends. In public ones, however, it was. This is where the cognomen or agnomen comes in - an additional name added to the fundamental two to serve as a diacritic.^(3) However, they originated in the aristocracy, and so were eagerly taken up as a means of showing your descent from a great man - so the cognomen 'Scipio' might have originally marked out a particularly distinguished (though now totally obscure) man in the gens Cornelia, but by being adopted by all of his descendants soon ended up being shared by every member of an entire (large) branch of a (very large) clan. So at that point you need to go further, and you end up with more nicknames being added - a particularly famous member of the clan was distinguished for his military service in against the Carthaginians in Africa, and so became *'*P. Cornelius Scipio Africanus'. Of course, his son then wanted everyone to know about his famous father, and so is also known as 'P. Cornelius Scipio Africanus', and his adopted son was originally a member of the gens Aemilia, but took his adopted father's name as a sign of respect, and therefore was known as 'Publius Cornelius Scipio Aemilianus Africanus'... you can see how it could get silly. If you really needed to tell people apart, say in an inscription, you could list their parents and grandparents - so one member of the same family is known on his tomb as 'Lucius Cornelius, Lucii filius, Publii nepos Scipio' - Lucius Cornelius Scipio, son of Lucius [Cornelius Scipio], grandson of Publius [Cornelius Scipio].

This is a long-winded way of saying that Roman names got very long indeed. In most cases, people were simply known by their nomen or cognomen, with the praenomen to distinguish if you really needed it (so in another poem, Catullus talks about his friend 'Cinna', realises that his interlocutor might not know who he means, and adds on 'Gaius'), or a note about who they were. But what you gave as 'your name' depended on who you were talking to (or writing for), and the situation you were in.