Why did Rome allow Jewish authorities to execute their own people?
Rome allowed Jewish authorities to execute their own people because Rome respected and condescended to Judaism as an ancient religion, granting it special permission to maintain temple authority and capital judgment. Rome even guaranteed safe delivery of temple offerings. This tolerance did not extend to Gentiles or to Christianity, which Rome viewed as new and therefore persecuted.
Les explains that Rome tolerated Judaism as an ancient religion and even condescended to it. The Jews were allowed to maintain the authority needed to keep the temple going. Jews in the dispersion sent substantial sums back to the temple, and Rome never intercepted those funds; Rome actually guaranteed safe delivery of those offerings. Because of this respect for ancient religions, Rome gave Jewish authorities permission to put their own people to death. That is why Paul could say he shut up saints in prison and voted to put them to death, while Rome would never have allowed such treatment of a Gentile. The same tolerance did not apply to Christianity, which was new rather than ancient; Rome tried every way to stamp it out, so Christians suffered massive persecution under Rome, while the Jews did not.