It depends on the timeframe and context.

Executions that were mandated by the state were overseen by the tresviri capitales, which was a college of three men who were, among other things, tasked with overseeing executions. This is probably the most like the professional executioner of later times.

In Roman law the lines between public and private law were different from today: some capital crimes were part of private law, which means the aggrieved party had to execute the condemned themselves. During the late Republic, most capital cases shifted from private to public law, with the installation of the quaestiones. The good news for Roman citizens is that for them, capital punishment was exceedingly rare by that time, even in most cases of murder.

Capital punishment for non-citizens was much more prevalent, and their executions were carried out by a large range of people, and even animals. Non-citizens in the provinces were at the mercy of governors, who could mete out the death penalty almost at will. Non-citizens in Rome had fewer rights than citizens, and slaves even had next to no rights. In those cases, the death penalty was usually carried out by soldiers, although the condemned might have ended up in the arena, fighting as gladiators or against the beasts, in which case it is hard to point out an executioner. The same goes for people who were condemned to the mines, which was pretty much a death penalty.