We consider a wireless network with distributed processing capabilities for estimation or detection applications. Due to limited communication resources, the network selects only a subset of sensor measurements for estimation or detection as long as the resulting fidelity is tolerable. We present a distributed sampling scheme based on the concept of innovations diffusion to select the sensor nodes. In the proposed scheme, sensor selection is accomplished through local communication and signal processing. In order to conserve energy and prolong system lifetime, the proposed algorithm selects a nearly minimum number of active sensors to ensure a desired fidelity for each working period. Extensive simulations illustrate the effectiveness of the proposed sampling scheme.