Not quite so, Pilate "hands over" Jesus to the Jews to be crucified. He never sentences him and "washes his hands" of the matter, signifying the abandonment of secular authority. The Jews (here really just a faction of the Jews, particularly the Sadducees) have already admitted that they can't kill him under the Jewish law but do so anyway. So both are guilty and the old law has been abandoned, represented by the curtain in the temple ripping open on Jesus' death.
Not quite so, Pilate "hands over" Jesus to the Jews to be crucified. He never sentences him and "washes his hands" of the matter, signifying the abandonment of secular authority. The Jews (here really just a faction of the Jews, particularly the Sadducees) have already admitted that they can't kill him under the Jewish law but do so anyway. So both are guilty and the old law has been abandoned, represented by the curtain in the temple ripping open on Jesus' death.
The Sadducees were just Roman collaborators who did what they were told by their Roman masters, so in the end, it was the Romans who executed Jesus.