Helen Bond presented a nice concise summary of the state of this question and evidence in her 1994 thesis. You can also follow up on her references for more detail, but for now, see Pontius Pilate in History and Interpretation, pp 27-28:
Quote:The supreme judicial competence of the prefect or procurator of Judaea to execute capital sentences was linked in all probability with the curtailment of the rights of Jewish courts to execute capital sentences. Although the evidence from Judaea is not entirely conclusive and the precise competence of the highest Jewish courts is still a hotly debated issue(1), the prevailing scholarly opinion is that Judaean courts did not have the right to carry out a capital sentence. This would have prevented native courts from eliminating the leaders of pro-Roman factions. Sherwin-White argues that the only places which were allowed to retain the right of capital punishment were highly privileged communities known as civitates libertae, ‘communities which for past services to the Roman state were made independent of the authority of the Roman magistrates in local administration and enjoyed unrestricted jurisdiction over their own citizens’(2). No such extraordinary concessions were likely to be found in Judaea. The Jewish authorities may however have been able to convene a court to discuss a capital case and even to reach a verdict, but the final decision seems to have rested with the governor. The gospel accounts of the execution of Jesus of Nazareth back this up, indicating that after a preliminary hearing by a Jewish court the case had to go before the governor.
(1) Evidence generally brought in favour of the Jewish courts having capital powers under the Romans are: the rule concerning Gentile trespassers in the temple (War 6.124-226, Antiq 15.417; also confirmed by an inscription, RA 23 (1882) p 220) which may be an extraordinary concession; the execution of any Jew trespassing in the Holy of Holies (Philo, Leg 307) which may be simply hypothetical; the death of James (AD 62, Antiq 220.200-203) which occurred at a time when Judaea had no governor and may have been illegal; and the death of Stephen (Acts 6:12-15,7:57-59) which may have occurred during the reign of Agrippa I (AD 41-44). Winter suggests that the penalty of strangulation was introduced after the deprivation of the Jewish right to judge capital matters in AD 70 (Trial, pp 67-74); however, this theory has not been proved.
Arguments in favour of the curtailment of Jewish powers of capital punishment are: the statement of Jn 18:31 (though this could be theologically motivated); the gospel accounts generally; and several rabbinic sources, though it must be admitted that these can only be used with care – Mcgillat Taanit 16, Mekhiltas of R. Ishmael, Simeon ben Yohai on Ex 21:14, jSanh 1,1/18a, jSanh 7,2/24b, Avona Zarah 8b, Sanh 4/a. For a fuller analysis of these rabbinic texts see Lemonon, Pilate, pp 81-90. See more generally Catchpole, Trial, pp 236-234; Lohse, TDNT, 7 p 865; Kilpatrick, Trial.
(2) Such as that on Rhodes (which subsequently lost the right after crucifying a Roman citizen), Dio 60.24.4; Sherwin-White, Roman, p 37.