This is how the chain of command in Judaea Province went.
In Jerusalem, on an everyday basis, Jewish leaders such as the high priest and his advisory council (
synedrion in Greek - the ‘Sanhedrin’) and the aristocrats of the city - many of whom were also priests - were responsible for running the government in place of the Roman prefect. It was the Temple guards who had regular police duties; the prefect’s auxiliaries in the Antonia and various other forts across the country seldom had police duties.
This is pretty much why it was the chief priests and the Temple police that arrested Jesus. Normally, the local police force would be the ones who would go about arresting any criminals or troublemakers and handing them over to the chief priests and the leading citizens, who were the responsible officials of the city, who would examine cases on behalf of the Roman governor.
Very serious cases would have been passed on to the prefect (because after all, the death penalty was officially in the hands of the Romans), although it’s also possible that at certain times the Jewish leaders might have taken care of such dangerous criminals by themselves by performing summary executions of their own - which would technically be illegal. But it’s not as if the Romans were totally able to police the internal life of the provinces closely; usually the Romans just winked at these illegal executions (hey, it reduced the number of cases
they had to review

) as long as the Jewish officials made sure that they aren’t discovered doing it.
The high priest Ananus/Ananias, son of Ananus/Ananias/Annas - the one in the gospels, the father-in-law of Caiaphas - was deposed in AD 63
precisely because he took advantage of the time in between (name removed by moderator)ial assignments - that interim period when the old governor left Judaea and the new one still hadn’t come yet - to violate the exclusion of the death penalty by ordering the execution of James the Just, the ‘brother’ of Jesus. He was deposed because his actions caused a
public uproar; it got too much publicity.
The stoning of St. Stephen (which is explicitly a mob lynching; there was no verdict and no sentence involved in the stoning, which as Luke describes it, was the result of the council getting so annoyed and angry at Stephen) and Saul/Paul’s persecution of Christians under authorization of the high priest could also represent times when Jewish officials bent the rules. In fact, these incidents might have happened - just like the execution of James - during the interim period when there was no Roman governor present in Judaea.
It was when the local Jewish leaders and the Temple police could not handle a situation on their own that the prefect and his 3,000 auxiliaries would actively step in. In turn, it was when the governor and his troops could not handle a situation on his own that his superior, the legate of Syria, and the legions he commanded would step in. So it’s like a chain of command: the Jewish leaders and the native police force at the bottom, above whom was the prefect and the auxiliaries, above whom was the Syrian legate and the legions.
As far as we know, the legions in Syria never went down to Judaea during the time of Jesus. All the recorded incidents Pilate was involved in (
the standards/shields,
the aqueduct riot,
the Samaritan prophet) seem to have involved his own auxiliaries. Pilate apparently never had to call on the legate for assistance. Indeed, that the legions came during the Jewish-Roman War thirty years later is an indicator of just how the situation in Judaea had deteriorated at that time.
Meanwhile, this is how the situation in the Galilee went.
As mentioned earlier, Herod Antipas the tetrarch had his own army with which to defend his tetrarchy. His soldiers would have been a combination of Jews, local non-Jews and half-Jews (Samaritans, Idumaeans, Syrians) and foreigners - as were most armies of the day, such as his father’s. (Herod the Great for example employed Gauls, Thracians and Germans for his personal bodyguard.)
Unlike the Jewish leaders in Judaea, Antipas as client ruler served as the representative of the Roman Empire, and was therefore vested with its capital power. That’s why he was able to execute John the Baptist and also sought to kill Jesus (Luke 13:31) - he could legally do so.
As client ruler, Antipas had to contribute to the Empire - who served as the patron in this
patron-client relationship - in some way. Two of the conditions he had to observe as client was that he had to defend his own borders and not allow revolt at home (that’s why he had his own army

) and had to contribute military aid when Rome required him to do so. In exchange, Rome took care of its client by sending military aid should Antipas run into trouble that he couldn’t handle by himself.
And that’s precisely what happened one time: Antipas decided to marry his half-niece, Herodias, who was already married to another of her half-uncles - the relationship John the Baptist was highly critical about. The first wife (whose name is unknown) managed to avoid divorce by crossing the border to her father, Aretas, ruler of the Arab kingdom of Nabataea. Sometime later, Aretas took advantage of a border dispute to invade the Galilee and thrash Antipas’ army - which his subjects regarded as divine punishment for executing John the Baptist. After some slow communications, Roman troops were sent from Syria (which was where the nearest substantial body of Roman troops was) to punish Aretas for invading Rome’s ally. However, the emperor (Tiberius) died, orders were suspended, and Aretas managed to get away.