The educated Romans learned the Attic Greek of Athens which was very difficult and also they and the normal Roman would know the more popular Koine Greek.
If a Roman was wealthy enough his wetnurse was probably Greek and those were the first words he heard.
People don't know this because of the Shakespeare play but Julius Caesar did not say in Latin "Et Tu, Brute?" when Suetonius said Caesar's last words were "Kai sy, teknon?" which is the Greek for "You too, descendent?" or "You too, child?" (It was common gossip in Rome that Brutus was Caesar's illegitemate child).
So in that case you can see that while Latin was used for formal functions, Greek was the language the Romans used for intimate purposes between themselves.