If you were a magistrate who held imperium then you would have bodyguards called Lictors. The number varied by the office that a person held. So for example a consul (one of the two chief executives of the Roman republic) had 12 while a Quaestor (a relatively junior financial official) had only one. They were originally plebs but later became freedmen or cirizens and were often retired centurions. While outside Rome the lictors carried fasces (a large bundle of sticks) with an axe head embedded in it, within the pomerium however the axes were removed - essentially meaning that they were armed with clubs. However the lictors of dictators kept them even within the pomerium. Their function was to protect their office holder. If he was in a crowd they would beat the people back to ensure that he could walk through safely and they accompanied him wherever he went including his home and the baths.

Quite aside from the lictors, most prominent romans were never alone in public. An important senator when he went to the forum to campaign for election for instance would be surrounded by a large group of his citizen clients. People further down the social scale might take their freedmen and slaves. The larger and more socially the group the higher the status of the individual. While these people were not bodyguards per se the fact of this massive crush of people around an important person must have exercised some sort of protective function.

There's a lot to say here but you should definitely look at /u/XenophontheAthenian's answers to:

Before the formation of the Praetorian Guard, who would Guard and Protect the Senators, Consuls etc. in Rome?

How did republican Rome function without a police force?

Did retired gladiators ever act as bodyguards to Roman politicians