i would assume anyone that’s using a gamma beam to communicate would understand, and compensate for, stellar drift during the communications time
of course, gamma radiation would be too slow to get there in any reasonable time frame.
which would leave you with entangled particles as the primary mode of communication... something we could never eavesdrop on or find evidence of
Entangled particles also have the advantage of providing no time delay which is important at interstellar distances.