There’s two possible ideas. One is that Jews did have limited right to give out death sentences, but only did so in the traditional methods (stoning, beheading, burning, and strangulating) and mostly in religious matters; secular cases being tried under Roman law. The other is that Rome had reserved the right of capital punishment to itself. I personally think the latter is more likely, since in other areas Rome usually withdrew the right of capital punishment or gave only limited use of it in order to prevent local courts from legally turning against sympathizers and collaborators to Roman rule. A decree of Augustus to the proconsul of Cyrene, dated 7-6 BC, which made provision for provincial courts to practice “delegated jurisdiction except in capital cases,” which are to be kept in the hands of the proconsul himself, shows just how picky the Romans were about restricting the power.
During the 1st century, the provincial ruler’s power was very considerable; it was only in the following century that authority was severely limited by the emperor himself. In these outposts of the empire, the prefect had to be able to do whatever he thought necessary for the good of Rome, and this included the power to discipline the army. The prefect’s right to sentence people to death was not only exclusive but also absolute; he could execute a citizen, and he did not have to formulate a charge that would stand up in a court at Rome.
Whichever camp you take, the only exception to this rule that we know of
is when a non-Jew trespasses into the ‘Jews-only’ part of the Temple (non-Jews were forbidden to enter anything other than the outermost parts of Herod’s temple; the Court of the Gentiles was the furthest they could go): “No outsider shall enter the protective enclosure around the sanctuary. And whoever is caught will only have himself to blame for the ensuing death.” This is keeping in line with the Roman custom of granting provincial subjects as much freedom as possible in practicing their religion, and besides, the Jews would wish to maintain the purity and sanctity of the Temple hence, the special exception.
Assuming that the second possibility is correct, what are we then to make for Stephen’s death? One could argue that the stoning of Stephen is not so much an official execution as a lynching: at least as Luke records it, there was no verdict and no sentence (Acts 7:57-58). We need not go to lengths to assume as some people did that the passage of the death sentence was omitted by a copyist. Which goes to the question: why didn’t the Romans step in to stop this? The answer is simply because the Romans could not police the internal life of the provinces closely, and even if they could, they would have simply batted a blind eye. In other words, provincials frequently took matters into their own hands and illegally executed natives on criminal charges, but government officials simply ignored them - as long as Roman citizens were not involved.
To put things into perspective, Judaea was a province ruled by a prefect (or after AD 41, a procurator), who commanded auxiliaries recruited from
peregrini, local non-citizens: being an equestrian, he was not of high rank enough to command a legion; his immediate superior, the legate of Syria, meanwhile, had four legions under his belt. And even then, unlike in standard retellings of the Jesus story where Romans are barbarians who roam, loot and pillage around the countryside, all the while kicking old men and forcing people to carry their loads, in reality the prefect and the majority of his soldiers stayed in the capital of Caesarea Maritima for most of the year far away from the sight of Jews, preferring instead to let native magistrates (in the case of Jerusalem, the high priest and his advisory council) run daily affairs for him. These locals were the responsible officials for the area and was normally in charge of ordinary police and judicial procedures, though the prefect would have the final word. All in all, the prefect only had 3,000 troops at his disposal, which is not sufficient to handle serious trouble, thus in emergencies he would need the aid of the Syrian legate and his legions.