Contemporary slave scholars can sometimes overestimate the importance of slaves being property in Greek and Roman minds. Personally, I’m of the opinion that this aspect of slavery was one that the Greeks and Romans did not dwell on very much. The master-slave relationship was more about power than it was about property.
This perspective also makes it easier to understand one of the functions of the Thesion. A number of authors, most extensively the antiquarian Pollux, explains that in fifth and fourth century Athens slaves could flee to this temple of Theseus and plea for mercy. Mercy did not take the form of freedom but rather their sale to another master. It is unclear how exactly this worked, but Christensen points to the role of the slave’s advocate for some possibilities. Only Athenian citizens had standing in court and had the authority to file cases. As a result of this, along with a known law that makes it illegal to act maliciously towards slaves, Christensen argues that an advocate could represent the slave in court against the master from whom the slave had fled. Perhaps Athenians were incited to advocate for slaves either because of the exemplum of Theseus (after whom the Theseion was named) or perhaps if they successfully defended the slave they had the opportunity to buy the slave.
In any case, I have a very hard time imagining any sort of similar system arising in Rome. While both the Greeks and Romans had hierarchal societies and did not understand the ideal of equal political power or access extending to everyone in a community, nonetheless the Romans appear to have been more sensitive about issues of rank. Greeks also commented on this. Dionysius of Halicarnassus argued that the Romans, out all of the people in the Mediterranean, had the most control over their sons.