The person to read for this is Christopher Zeichmann. His dissertation, revised and published as The Roman Army and the New Testament, goes into great detail both about the history of the military in Roman Judea and Galilee, and also about the way the military is portrayed in various New Testament books. I believe he also has a website that compiles the relevant epigraphic evidence.

To do Zeichmann no justice, before the first Judean-Roman War (ca. 66 CE), there were no legions in the area, though there were up in Syria. Herod's army, as well as those of his successors, comprised mostly ethnic Syrians, Judeans, and Samaritans, who seem to have been recruited from the Herodian cities of Sebaste and Caesaria Maritima. Josephus mentions Herodian soldiers from as far away as Germany, but possibly only brings them up because they were out of the ordinary. So, before the war, and therefore during Jesus's lifetime, the majority of soldiers in Judea and surrounding areas would have been locals, and spoke dialects of Aramaic, and possibly also Greek. The legions arrived in and after 66, most famously legio X Fretensis, which eventually garrisoned in Judea/Syria Palaestina until at least the third century. Fretensis was more multiethnic than the previous auxiliaries and royal armies, but still mostly comprised easterners, who, again, would have primarily spoken various dialects of Greek and Aramaic, although we see more Latin inscriptions after the war. Centurions and higher ranking officers would have needed Latin familiarity to communicate with their superiors, but weren't necessarily Italian. Studies from Roman Britain suggest that imported soldiers often retained ethnic identity including language, even in multicultural units (I'm sorry, I don't have the reference right now because I'm away from my notes, but if you really want it DM me and I'll happily hunt it down next week), so it's reasonable to suppose the same of eastern units, even when we see evidence of Judean or other non-Italian soldiers taking Latin names and naming conventions.

This is all historical detail, gleaned mostly from Josephus and archaeological/epigraphic evidence. For understanding the Gospels and thus Passion, though, it's important to recognize that the people of Judea experienced the arrival of the legions as the arrival of Roman soldiers, regardless of ethnicity. The Gospels were all written after 70 and, in their various ways, represent responses to or at least developments from the war. Therefore, the portrayal of soldiers and especially higher-ranking soldiers in NT texts tends to identify them as Roman legionaries, even though for the period they're depicted (ca. 30 CE), there were no legions and most soldiers were locals.