In the gospel accounts of the trial of Jesus we learn that the Jewish authorities could not sentence anyone to death:
Pilate said, “Take him yourselves and judge him by your own law.” “But we have no right to execute anyone,” they objected. (John 18:31 NIV)
But it's easy to get confused, because the citizens of Jerusalem were later stoning Stephen to death:
When the members of the Sanhedrin heard this, they were furious and gnashed their teeth at him. But Stephen, full of the Holy Spirit, looked up to heaven and saw the glory of God, and Jesus standing at the right hand of God. “Look,” he said, “I see heaven open and the Son of Man standing at the right hand of God.” At this they covered their ears and, yelling at the top of their voices, they all rushed at him, dragged him out of the city and began to stone him. Meanwhile, the witnesses laid their coats at the feet of a young man named Saul. While they were stoning him, Stephen prayed, “Lord Jesus, receive my spirit.” Then he fell on his knees and cried out, “Lord, do not hold this sin against them.” When he had said this, he fell asleep. (Acts 7:54-60 NIV)
You might argue that they were not authorized to do this but they did it out of pure rage. But they had enough presence of mind to put their coats at the feet of Saul (there is another question about this matter specifically).
Are these texts reconcilable against what we know of contemporary legal systems? Did the Jewish authorities of the time have the power to order executions according to their own laws and the Roman rule they were subject to?