Let us consider the 4 most prominently spoken languages in that region at that time.
The lingua-franca of the Eastern Roman empire, it had been widespread in the East since the time of Alexander. The prevalence of a common language was a very effective tool for Roman governance. Greek was the language Rome used to communicate (in the east) with the people it subjugated (source).
Pilate, an educated man and a member of the equestrian order (source), would most definitely have spoken Greek.
For the evidence that Jesus would have spoken Greek, see:
- Stanley Porter's work here
- JN Sevenster's work here
- My video here: What Languages Did Jesus Speak?
- The prevalence of Greek speaking Jews in the book of Acts (including Galileans with Greek names)
- Jesus' time spent in the Greek-speaking cities of the Decapolis
Pilate & Jesus both spoke Greek, and Pilate would have expected to communicate with his subjects in Greek. This gives a high probability to their conversation taking place in Greek.
The familial language of many Jews as a result of the Babylonian captivity, Jesus undoubtedly spoke Aramaic (and several of his statements, in Aramaic, are preserved in the Gospels).
Pilate was a transplant to the region and there is no evidence he ever spent any significant time in the far east. It is unlikely that he spoke Aramaic, nor would he as a Roman official have felt it his duty to learn the common-tongue of the people he ruled.
Jesus could have spoken Aramaic to Pilate, but the conversation would have had to have been conducted through an interpreter.
The religious language of the Jews, and the language of the synagogue. Prior to the destruction of the temple, the Tanakh was not written in Aramaic (see my post here); the scrolls in the synagogues were Hebrew documents.
Jews in Judea & Galilee would have regularly encountered Hebrew as they worshipped and learned in the synagogue. Boys who received even the most rudimentary education would have been taught the Torah in Hebrew. Although Jesus may well have been familiar with the Septuagint (the Greek translation of the Tanakh), His primary association with the Jewish scriptures would have been in Hebrew.
A century ago it was fashionable to claim that Hebrew was a dead language by the first century, but the Dead Sea Scrolls have since undone these arguments. Josephus, Rabbinic literature, coins, and the New Testament itself also give evidence that Hebrew was still being spoken. See further discussion in this Hermeneutics post by Frank Luke.
- For evidence that ἑβραϊστί in the New Testament is a reference to Hebrew, not to Aramaic, see Buth & Pierce's work: Hebraisti in Ancient Texts: Does ἑβραϊστί Ever Mean 'Aramaic'?.
- For a review of the political (and anti-Semitic) reasons for which 19th-century German scholars tried to convince the world that Jesus did not speak Hebrew, see Baltes' work here.
Jesus spoke Hebrew. There is no reason to believe that Pilate, a man antagonistic towards the Jews (source), ever learned the Hebrew language.
Latin was the original language of the Romans and remained the language of imperial administration, legislation, and the military throughout the classical period (source)
Pilate spoke Latin, as did many of the soldiers at his command. Latin was not the lingua franca of the Eastern Empire and was not what Romans used to communicate with their subjugated peoples in the region. There is no reason at all to believe Jews in the region would have taken the time (or had the opportunity!) to learn the administrative language of their pagan overlords.
Pilate had a sign posted (see Luke 23:38) at Jesus' crucifixion in Latin (Rome's language), Greek (the lingua franca), and Hebrew (the Jews' language--note that Aramaic was spoken by the Jews, but the most important writing was done in Hebrew).
Perhaps the plainest (and admittedly ever-so-slightly over-simplified) rendering that I have encountered of the trilingual nature of the Jewish world in which Jesus lived is: Aramaic was the language of the home, Hebrew was the language of the synagogue, and Greek was the language of the marketplace. (Latin being a language of soldiers & politicians).
- Pilate spoke Latin & Greek.
- Jesus spoke Aramaic, Hebrew, and Greek.
Although this does not rule out the slim possibility that they communicated via an interpreter, in none of Jesus' or the apostles' conversations with Roman officials is there any hint that an interpreter was needed.
The most practical language for them to use in conversation was Greek.