In a modern ATC system the controller's display shows a difference between "own" and "other" traffic
Well lets all hope the system is modern! There are a couple of waypoints off the northern coast of Spain where the Spanish regularly forget to handover crews to the French, resulting in the Spanish calling an aircraft thats out of VHF range and the French trying to contact it on 121.5. It became so frequent that I always made a point of calling the Spanish for a "radio check" every time I approached these waypoints to remind them to hand me over.