Here we elucidate a paradox: how a single chemoattractant-receptor system in two individuals is used for communication despite the seeming inevitability of self-excitation. In the filamentous fungus Neurospora crassa, genetically identical cells that produce the same chemoattractant fuse via the homing of individual cell protrusions toward each other. This is achieved via a recently described "ping-pong" pulsatile communication. Using a generic activator-inhibitor model of excitable behavior, we demonstrate that the pulse exchange can be fully understood in terms of two excitable systems locked into a stable oscillatory pattern of mutual excitation. The most puzzling properties of this communication are the sudden onset of oscillations with final amplitude, and the absence of seemingly inevitable self-excitation. We show that these properties result directly from both the excitability threshold and refractory period characteristic of excitable systems. Our model suggests possible molecular mechanisms for the ping-pong communication.