Did Rome allow the Jewish people to carry out their own religious capital punishment?
Yes. Rome allowed the Jewish people to carry out capital punishment in the name of their religion, with no opposition at all. This explains why Paul received authority from the chief priests, not from 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, and there was no opposition to that at all. He notes that Paul received authority not from Caesar or Rome but from the chief priests to imprison many Jewish saints. The context then asks why the Jews demanded Rome put Christ to death. Les answers that this was prophecy: both Jew and Gentile would put the Messiah to death, and it could not be by the Jew alone. Scripture also required a lifting up or crucifixion, as with the serpent in the wilderness and John chapter 3, so stoning would not have fulfilled those prophecies. Thus it was Jew and Gentile who carried it out.