I heard in a sermon that it was not allowed for the Jewish leaders to convict a person without permission from the Roman governor. My question is: how did they stone Stephen (Acts 7:58) without a trial in front of the Roman governor?
The Bible does not reveal the the legalities or illegalities of all the killings it records. From the scanty histories available, most scholars seem to think that at the time of Pontius Pilate, only the Roman governor had the authority to order capital punishment . We must remember, however, people back then did not enjoy the equal protection of the law we enjoy today in a democratic nation.
Politically, Rome would not have had any interest in this case, so the murderous mob may have had little to fear. On the other hand, many people knew and admired Jesus, so there may have been some political motivation for Rome to take notice of this case and assume the duty of legal arbitration. From Pilate's perspective, however, he may have thought that in this matter, being forgiven for an error in judgment was better than obtaining permission to make a judgment. Then, too, the crowd in its demonic hate gave Pilate a not-so-veiled threat,
"'If you release this Man, you are no friend of Caesar; everyone who makes himself out to be king opposes Caesar'" (John 19:12).
By referring to comments by Josephus, some of the better commentaries try to work out a scenario in which the lack of interest by Rome was only natural in this instance. From the Baker New Testament Commentary, p280, we read,
The question of the legality of Stephen’s death is a difficult one. By all appearances, his death is the result of mob action that the Romans did not prevent. Nevertheless, the Roman governor could conduct an inquiry, because the Jews might not administer capital punishment (John 18:31) . . .. That power belonged to the Roman governor. To illustrate: Josephus states that the Roman procurator (governor) Coponius, sent to Judea by the emperor, was “entrusted by Augustus with full powers, including the infliction of capital punishment.”
If we assume that Stephen died in A.D. 35, Pontius Pilate was still the governor of Judea. At that time Pilate’s troubles, resulting from his slaughter of numerous Samaritans at Mount Gerizim, were sufficient evidence to demand his recall (A.D. 36) at the request of Syria’s governor. . . . In this political climate, the Jews would not fear repercussions for killing Stephen. To be sure, in his last year in office Pilate had lost influence and authority in Judea.
Stephen was not killed pursuant to a lawful trial and execution. He was murdered by a mob. Mobs pretty much by definition don't obey the law. Just because something is illegal doesn't mean that nobody ever does it. If that was true, there would be no need for such things as prisons.
To the best of my knowledge, neither the Bible nor Fox's Book of Martyrs nor any other history book I know of mentions any action by the Romans against those responsible. Of course the Jewish authorities weren't going to punish anyone as they were in favor of the whole thing. Maybe the Romans did nothing because they didn't want to create a conflict over one religious fanatic.
Whether or not the Jews were allowed to stone without direct Roman approval in each case is the question. Where does the idea that they were not allowed by the Romans to do this come from?
John 18:31-32  Then said Pilate unto them, "Take ye him, and judge him according to your law." The Jews therefore said unto him, "It is not lawful for us to put any man to death":  that the saying of Jesus might be fulfilled, which he spake, signifying what death he should die.
In the gospel of John this is what the Jewish leaders say to Pilate. But were they being entirely truthful? And what did they even mean by this?
Traditionally preachers have interpreted this as a legal statement conceding that under Roman rule they don't have the authority to execute anyone without Roman approval. But if that were the meaning, then surely Pilate had just given them the approval!
"Take ye him, and judge him according to your law." So the real meaning is as the passage says, they said this because they wanted him crucified (not stoned). It is fairly clear that John intends for us to comprehend that the Jewish leaders are lying here, and that Pilate lets them get away with the lie, and that the reason Pilate ignores their lie is that divine providence is at work for God wants Jesus' saying specifying crucifixion as his means of death to come true.