A couple thoughts. I hear very often that Jesus did not speak Latin. Considering Jesus is divine, lived in a Roman province, and interacted with Romans, He probably could speak Latin (even if He never once spoke it) and certainly did understand it. For Romans Latin would’ve been their primary language, though they likely learned Greek first. Officials would’ve likely known Aramaic as well. But I digress…
I doubt that there would have been
regular interactions between Jesus and the Romans. For one, unlike in
certain old movies where Roman soldiers perpetually patrol the streets and bully locals and, well, just be plain menacing (a picture which would probably be more accurate during the Jewish revolt of the 60s rather than the AD 20s-30s), at our time period Romans usually contented themselves with keeping themselves away from sight.
In Judaea, the prefect and most of his troops - which, considering that the prefect was of
equestrian status and thus not of high rank enough to command a legion,
would have been composed of 3,000
auxiliaries recruited from local subjects who did not hold Roman citizenship (
peregrini), usually from areas like Samaria or Syria - would have stayed in Caesarea Maritima, the capital of the province, with other gentiles for most of the year. Since Rome did not govern Judaea on a day-to-day basis but apparently preferred an indirect form of rule (hey, it was effective), the prefect’s normal duties, such as management of daily affairs and collection of taxes, were usually handed down to local officials, such as the high priest and his council. The prefect would have only showed up a few times each year in public, usually in potentially dangerous seasons like Passover to watch out for potential unrest, to ensure that the huge crowds did not get out of hand, hence the reason why Pilate was ‘available’ in Jerusalem during Good Friday. Aside from such special occasions, regular Roman presence was largely limited to the small garrison in the Antonia Fortress just beside the Temple (because you can’t completely trust those Jews, I guess

) and certain other fortresses in the countryside.
In the Galilee, meanwhile, the tetrarch Herod Antipas was pretty much independent. Rome still had the final say on crucial matters, but he was more or less free to run the kingdom as he wished. He fulfilled the main conditions by keeping order, paying tribute and cooperating with Rome, and in return Rome gave him protection against implied threat of retaliation against invaders. The troops and the taxes were his (though Rome would have supplied him with additional reinforcements when necessary), district governors were his appointees, and local rulers served at his pleasure. Hence, there would have been no major Roman presence.
I quote from E.P. Sander’s
The Historical Figure of Jesus:
Since so many readers and scholars of the New Testament have imagined Jesus living in a context in which the populace was daily oppressed by Roman soldiers and officials, and some scholars are now claiming that there was a substantial Gentile population in the Jewish cities, I wish to re-emphasize the actual situation. In Galilee there was no official Roman presence at all. Greek-speaking Gentiles lived in the cities that they had long inhabited, which formed a kind of crescent around Galilee: there were Gentile cities to the east, to the north and to the west. In geographical Galilee there was one Gentile city, Scythopolis, but it was independent of political Galilee. In Judawa the official Roman presence was very small. There was one Roman of rank in residence, and he was supported by a handful of troops. This Roman and his small military force lived among a lot of other Gentiles in Caesarea, seldom came to Jerusalem, and did nothing to plant Graeco-Roman laws and customs to the Jewish parts of the country. In Jerusalem there was not a substantial Gentile presence. In the place of the image of Roman soldiers patrolling the towns and villages of Palestine, forcing Jews to carry their burdens, and motivating shopkeepers and farmers to cater to their tastes, we should think of a few Roman soldiers, banded together, living in or around one city, with only very small outposts occupying forts in potentially hostile territory. Effective rule was in the hands of local aristocrats and rulers.
Granted, there was the incident between Jesus and the centurion, but we should really imagine it as a kind of exception to the norm.
And you do raise a really intriguing question there. Did Jesus allow His divinity to ‘intrude’ into the normal aspects of His human life (say, His health, language, etc.), and if so, just how did it manifest itself?