The problem of detecting transient signals in noise under missing signal observations (samples) is considered. Specifically, a fusion center tries to detect the presence of a decaying signal in additive white Gaussian noise (AWGN) by collecting samples from distributed sensors through erasure channels, during which some of the samples may be lost. Under Neyman-Pearson detection, it is shown that missing samples cause performance degradation by reducing the signal energy received at the fusion center. Based on the assumption that the fusion center can control the sampling procedure through a feedback channel, an adaptive sampling policy is proposed with the goal of achieving accurate and timely detection with the minimum communication cost. The proposed policy is efficient and flexible in that it can be configured to yield a range of performance-cost combinations, where approximated closed-form solutions are derived for the configuration. Simulations show that compared with fixed-rate sampling, the proposed policy achieves significantly better tradeoff between detection performance and communication cost.