Strictly speaking, the whole region were Roman provinces under control of Romans. So by law, the Jews were not allowed to execute people.
However, the enforcement of the law is a different matter. Those were different times, the Roman Empire was not a police state in the modern sense, and there was a lot of ‘looking the other way’ when Jews killed other Jews, even if they were Christians. That explains the near stoning of the adulteress, the stoning of St Stephen, the beheading of St James the Greater, and numerous other martyrs. With Jesus it was a different story because he had broad-based support among Jews (at least at that point in time), so the Jewish authorities had to involve the Romans. With St Paul, again they had to involve the Romans because he was a Roman citizen.