Local law was very much allowed in the Roman Empire, as a holdover from Republican days, whether the matter was civil or generally criminal. Governors of provinces generally allowed local courts to handle legal matters, only stepping in when cases were referred to them; at this point, generally the local law would apply to provincials, while Roman law declared by the urban praetor applied to Roman citizens living there (John Richardson's summary article on Roman law in the provinces from the Cambridge Companion to Roman Law is a relatively readable summary for an academic text). As Richardson states, local law was eroded as the Empire continued.

However, the issue becomes thornier because you're specifically asking about criminal sentencing to death. This power - known as the ius gladii - was one reserved for the provincial governor (although this was not widespread in the first years of the Empire, instead remaining with the Emperor himself). However, this is not a totally clear-cut situation, as can be seen in a well known example - the sentencing of Jesus Christ.

John 18 states that the Sanhedrin - the Jewish court - go to Pontius Pilate and tell him that they do not have the power to sentence him to death despite him breaking a law that demands it (Leviticus 24 states that blasphemy is a capital offence, punishable by stoning to death). So far, so consistent. However, Pilate tells them to "take him (yourselves) and crucify him" and the Jewish historian Josephus recounts other instances of the Sanhedrin issuing and carrying out death sentences at the time. Smallwood's The Jews Under Roman Rule hypothesises that an exemption may have been made allowing the Sanhedrin the power of execution in matters of religion, but offers no proof of this, only the examples given above.