One of the most common rewards that Rome had to offer for services during its imperial and republic eras was citizenship and one of the major rewards for the allied auxiliary troops was citizenship for both them and their children at the end of the service term.
I suppose it wouldn't be hard for citizens to be identified in the early days of the republic, due to their somewhat homogenous appearance, however in the imperial days, we would likely have gauls, germans, greeks, iberians, egyptians, lybians and phoenicians holding the title of citizen, so in a situation like a census, how would a roman official go about determinating who was a citizen and who wasn't? Was there some sort of documentation that could prove it?
Freed slaves are over-represented in the very limited Roman census documents we have. They're also over-represented in funerary inscriptions (by "over-represented", I mean that the numbers appear to be high relative to the number of freed slaves in society. It's a fair assumption that the reason we're seeing more of these freed slaves than we might expect is that they were particularly eager to manifest their manumission in a tangible form.
Very rarely, we have surviving manumission documents, for example, this one from Roman Egypt, found in the Los Angeles County Museum, originally thought to be a "mummy label"
The tablet can be palaeographically dated to the second/third century and is of unknown provenance but clearly from Egypt, given its excellent state of preservation. It records the payment of the 5% tax on manumissions ( vicesima libertatis)6 and is a direct parallel to P.Mich. 7.462 (CPL 171; FIRA 3.10bis). As that parallel suggests, our text must have originally been written on two seal able tablets. Latin tablets were commonly used for Roman legal documents of all kinds. From Egypt a few survive, all relating - as our text does - to Roman citizenship.
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"(N.N. manumitted by N.N.) alias Helladios paid the 5% (manumission) tax to the state. I, Aurelius Isidorianos, agent for the procuratores of the 5% (manumission) tax have received the 5% manumission tax of the Roman people (from N.N.)."
While only a few such documents survive, they must have been very common-- this looks like boilerplate, effectively a receipt, albeit a receipt which would have been very, very important to the manumitted slave.
See The Regularity of Manumission at Rome for a comprehensive look at how manumission worked. We only have selective data, but there's reason to believe that it was very frequent, and that in at least some places "Roman slavery" may have looked more like what we'd call "indentured servitude"-- you'd be a slave until you could save up a sum to buy your freedom. There is famous comment by Cicero:
[A]fter a period of six years, during which we have been deprived of it, having endured slavery longer than prudent and industrious prisoners usually do,
M. Tullius Cicero, Orations, The fourteen orations against Marcus Antonius (Philippics)
. . . which he seems to be implying that a slave might expect manumission within six years. While there are arguments against taking this comment too literarily, its clear that Cicero, a wealthy man with many slaves himself, doesn't think of slavery as a lifelong condition, but rather a burden that you'd expect to endure and discharge.
So Roman slaves were frequently freed, and they and their families had both documents, and took care to record their manumission in more permanent media as well.