The Roman economy was heavily reliant on slaves, which means they had a LOT of them, which would lead one to conclude that at least some of them would run away. So who hunted escaped slaves? Did Rome have specialized people whose sole occupation was chasing and capturing escaped slaves?
It appears there were 'professional' slave hunters during Roman times, who were paid to hunt down and return fugative slaves.
From Johnston's The Private Life of Romans:
If he attempted flight, he must live the life of an outlaw, with organized bands of slave hunters on his track, with a reward offered for his return, and unspeakable tortures awaiting him as a warning for others. It is no wonder, then, that slaves sometimes sought rest from their labors by a voluntary death
So, yes, there an organized effort, including rewards, to retrieve escaped slaves. An article from BBC, Resisting Slavery in Ancient Rome, also mentions this:
To deal with the problem, the Romans hired professional slave-catchers to hunt down runaways, and posted advertisements in public places giving precise descriptions of fugitives and offering rewards for their capture. Around the necks of slaves who were recovered they also attached iron collars, giving instructions on what to do with the slaves who wore them if they happened to escape again. Examples can still be seen in museums.
One more source Ancient Rome: A Sourcebook, by Matthew Dillon, Lynda Garland, mentions these slave-hunters by name, fugitivarius, also lists them as a 'professional' group, being hired to retrieve the slaves, and specifically states there was no government organization to help in this recovery:
...it is quite clear from Cicero's account that there was no actual state apparatus for helping owners to recover their runaway slaves. The professional slave-catcher (fugitivarius, plural: fugitivarii) appeared in the late Republic to help slave-owners recover their property.
Johnston also mentions the difficulty in attempting to flee:
There was little chance of escape by flight. In Greece a slave might hope to cross the boundary of the little state in which he served, to find freedom and refuge under the protection of an adjoining power. But Italy had ceased to be cut up into hostile communities, and, should the slave by a miracle reach the border or the sea, no neighboring state would dare defend him or even hide him from his Roman master.