As a correction, the Egyptian languages spoken at the time of the existence of the early Roman Empire would have been Middle Egyptian as a liturgical and literary language and Demotic as the everyday vernacular. Koine Greek would have been spoken by the Ptolemies and by some administrators in their government, especially by those in the northern portion of the Nile delta.

You might get something out of Clarkson 2015 - Language and Society in the Greek and Roman Worlds.

For the part of your question which asks about how communication is conducted in environments when neither side shares a language, I highly recommend you to cross post it to the /r/linguistics weekly Q&A thread. You can also look at what Daniel Everett had to say about field work in such circumstances in a talk he gave for the Linguistic Society of America.

There are stories of Roman emperors meeting Pharaohs

A follow-up question on this, how did the pharaohs view roman emperors (and ancient Egyptians the Romans)?