As others have pointed out, there were plenty of polylingual people back then, and the Persians in particular would have had access to lots of subjects who were fluent in Greek, because they were Greek.

Another thing to recognize is that there were probably at least two languages that were widely understood because they belonged to people who engaged in lots of long-distance trade across the Mediterranean basin. One of those was, of course, Greek, since several wealthy Greek city-states (e.g., Athens, Corinth, Syracuse) did a lot of maritime trade. The other would have been Punic (Phoenician), and the home cities for that language were under Persian rule throughout the period.