The legal text Justinian's Institutes (1.3.3) gives an etymology for the Latin word for slave (servus) that is interesting for our purposes:

Slaves (servi) are so called because commanders order captives to be sold and so spare (servare) rather than kill them: they are also called mancipia because they are taken physically (manu capi) from the enemy. (trans. from Watson, 8)

This is not considered the actual etymology of these words, but it gives a sense of the importance of capture as a source of slaves. War captives would have been a huge source of slaves during the heights of military campaigns. Roman law allowed foreigners captured through war to be sold (but not citizens in civil war). So most people captured and sold into slavery were from the various groups conquered by Rome; there does not seem to have been much, if any, legal limit to capture of non-Romans (Watson, 20). The only mention is that people kidnapped by pirates or brigands continue to be free (Digest 49.15.19.2), but the enforcement of this rule would be difficult.

Citizenship would have been the important factor in your questions of proof and who could be kidnapped and sold. Before being sold, a kidnapped citizen or free person could have been ransomed by family or friends, instead. The wealthier or more influential the individual, the more likely this would be the case. If a free person did end up in slavery, it would take the testimony of another free person (like a friend or patron) to free them. It was easier to prove proof of ownership than proof of freedom. We have papyri surviving (mostly from Egypt) detailing writs of sale of slaves or slaves inherited via wills. These usually include details like name, age, and physical description in order to match the individual to the document.

This is not to say that most people kidnapped would be freed, as that was dependent on resources for ransom or connections. St. Augustine describes brigands kidnapping people near his home in North Africa and selling them overseas, where they would have little recourse to freedom (Letter 10; see also Bodel). Travelers, on land or sea, would have been especially vulnerable further away from home, and Suetonius describes slave prisons being full of travelers like its a common problem, saying that Tiberius undertook:

the investigation of the slave-prisons​ throughout Italy, the owners of which had gained a bad reputation; for they were charged with holding in durance not only travellers, but also those whom dread of military service had driven to such places of concealment. (Suet. Tib. 8; trans. Thayer)

Being captured and sold into slavery definitely occupied the Roman imagination as a possibility; free people being captured, sold, and redeemed was a common plot in Greek (and therefore Roman) comedies, and is also a plot point in Petronius's Satyricon novel. Romans were aware that very little separated slavery and freedom. That being said, it's debated how much such kidnappings contributed to the slave population, with most historians agreeing that it's a negligible amount compared to war captives or home-born slaves. There was a bit of a stigma surrounding slave traders (because they had a reputation for defrauding buyers, not because of the enslavement), so we have very little evidence for historical traders.

Basically, anyone part of a marginal and/or non-Roman group could be at risk of kidnapping and enslavement, but this was as not a major source of slaves as others.

Sources:

John Bodel. "Caveat emptor: towards a study of Roman slave traders." Journal of Roman Archaeology 18 (2005): 181-95.

W.V. Harris. "Demography, Geography, and the Sources of Roman Slaves." Journal of Roman Studies 89 (1999): 62-75.

Walter Scheidel. "The Roman Slave Supply." In The Cambridge World History of Slavery, Vol. 1. Edited by Keith Bradley and Paul Cartledge. Cambridge University Press, 2011.

Alan Watson. Roman Slave Law. Johns Hopkins University Press, 1987.