Did Rome allow Jews to execute people for religious reasons?
Yes. Rome permitted the Jewish people to carry out capital punishment in the name of their religion, and there was no opposition to it. That explains why Paul received authority from the chief priests, not Caesar or Rome, to imprison Jewish believers.
Les explains that Rome let the Jewish people carry out capital punishment in the name of their religion, with no opposition. This is why Paul, reflecting on his past, says he received authority not from Caesar or Rome but from the chief priests to shut up many of the saints in prison. Les then addresses why the Jews still demanded that Rome put Christ to death: prophecy had said both Jew and Gentile would put the Messiah to death, and it could not be by the Jew alone. Scripture also pointed to a lifting up, a crucifixion, as with the serpent in the wilderness and Jesus' words in John 3. Stoning would not have fulfilled those prophecies, and it had to be Jew and Gentile together who carried it out.