Without getting into all of the nuances in how Rome ruled the Empire, the underlying principle was driven by the concept of self-governance. Control from Rome limited to assuring allegiance, avoiding war, and paying taxes to Rome. As long as there was peace and taxes were paid, individual "kingdoms" within the Empire were allowed to govern themselves.
With regard to issues at work in Paul's rounding up Christians in Damascus, two other factors are present. First, on a purely personal level, a Roman citizen had higher legal rights than non-citizens, regardless of where they were in the Empire. For example, Paul exercised his rights as a Roman citizen and had his case transferred to Rome. The second factor is Judaism was a legal religion in the Empire. In the case of Paul going to Damascus, he was outside the legal jurisdiction of Judea and in that sense would be infringing on the self-governance in another "kingdom." However, anyone who claimed to be a Jew was subject to the rules of Judaism regardless of where in the Empire they were.
The Jews in Damascus, or in any location in the Empire, were subject to Jewish authority on religious matters. Since diaspora Jews recognized the authority of Jerusalem on religious questions, Paul's letter from the high priest gave him the legal right to arrest Jews who did not invoke their rights as Roman citizens and bring them to Jerusalem to be judged over the internal religious matters of Judaism.
Therefore, Paul's direct authority was from the high priest, whose legal authority emanated from the religion of Judaism. Roman authority was indirect in the sense Rome recognized Judaism as a legal religion and in so doing authorized another type of self governance for any individual who claimed to be a Jew.
Was Paul acting as a vigilante? No, he was acting legally as authorized by the high priest. He may have had a vigilante mindset and begged the high priest to "send me" not someone else. However, regardless of his motivation, he was acting legally under the authority as a Jew understood the Jewish law.
The involvement of Roman officials would be limited to ensuring the rights of Roman citizens were not ignored and perhaps secondarily, resolving issues where Paul attempted to arrest someone who claimed they were not a Jew and not subject to any Jewish authority. Although this second issue would more likely fall under the civil authority in Damascus, unless the individual was a Roman citizen.
Finally, it is important to remember the status of Judaism as a legal religion afforded Jews certain privileges other individuals did not enjoy. The ability to observe the Sabbath as a religious practice prevented military service and the requirement to worship only the God of Israel gave Jews protection from mandatory worship of local gods. On the individual level being subject to Jewish authority in Jerusalem is the other side of the coin of being a legal religion.
