It was Herod, himself, who seized Peter
And because he saw it pleased the Jews, he proceeded further to take Peter also.
Regarding the Jews concern about the guarding of Jesus' sepulchre after his burial (and prior to his resurrection) :
Pilate said unto them, Ye have a watch: go your way, make it as sure as ye can.
It is clear, therefore that the Roman occupiers permitted the Jews to have soldiery and a watch and to arrest and keep prisoners.
But Barabbas was held by the Romans because of the seriousness of his offences (sedition and murder). Luke 23:19.
It was the Jews who arrested Jesus and took him to Herod. But it was necessary to take Jesus to Pilate if they wanted (and they did want) the death penalty to be carried out. The Jews did not have the power to crucify.
But he who took Peter was Herod Antipas (the successor to the Herod who sent Jesus to Pilate) and he was a tetrarch - not a King - and his power came directly from Rome.
We know that Herod and Pilate were re-united in the aftermath to Jesus' condemnation and it may well be that this resulted in increased political confidence such that Rome 'promoted' either Herod or the succeeding Herod Antipas to a formal position and to Roman power.
So Herod Antipas did have the power to put to death and he used this power against James the brother of John.
I think it would be guesswork to surmise exactly how Jerusalem was being administrated (regarding imprisonment) during this period, considering Paul's interaction with the Chief Captain ('with a great sum obtained I this freedom' - the freedom of a Roman).
But whatever guards they were, Jewish or Roman, whom Antipas condemned to death, he most certainly had the power to do so.