Death penalty in the land of Israel in the Roman Empire

Were the Jews in the Second Temple Jewish era during the occupation of the Roman Empire were permitted to execute the death penalty without consent from the Roman government?

ב"ה Shalom The words of our sages in the Talmud are trustworthy and if it says in the Talmud which is part of our unbroken chain of tradition, that they carried out the death penalty, this is something to rely upon. All the best