How could Stephen be stoned without a trial and conviction in front of the Roman Governor? 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,

&quot;'If you release this Man, you are no friend of Caesar; everyone who makes himself out to be king opposes Caesar'&quot; (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.