This letter examines a variation of the canonical distributed detection system: a sensor may overhear other sensors' transmissions and thus may choose to refine its output in the hope of achieving a better detection performance. We show that while this is indeed possible for the fixed sample size test, asymptotically (in the number of samples) there is no performance gain, as measured by the Kullback-Leibler distance achievable at the fusion center, provided that the observations are conditionally independent. For conditionally dependent observations, however, we demonstrate that asymptotic detection performance may indeed be improved when overhearing is utilized.