Of course, this could just be me not understanding which languages were prevalent in which areas of the Roman Empire. (if someone does have info on that I'd like to know about it)
The elites from the Western half spoke Latin, the elites from the Eastern half spoke Greek. Indeed, most of the soldiers raised from the Eastern Empire would speak Greek and use Greek as their lingua franca, and colonial administrators would almost certainly be bilingual in Latin and Greek (as most heavily-educated Romans were). Though Pilatus would've definitely had Latin as his native tongue, he would've been fluent in Greek and been able to converse with the Temple Priests and other Jewish elites in it (though it'd be very doubtful he'd know any of the Aramaic of his subjects).