This paper considers an estimation network of many distributed sensors with a certain correlation structure. Due to limited communication resources, the network selects only a subset of sensor measurements for estimation as long as the resulting fidelity is tolerable. We present a distributed sampling and estimation framework based on innovations diffusion, within which the sensor selection and estimation are accomplished through local computation and communications between sensor nodes. In order to achieve energy efficiency, the proposed algorithm uses a greedy heuristics to select a nearly minimum number of active sensors in order to ensure the desired fidelity for each estimation period. Extensive simulations illustrate the effectiveness of the proposed sampling scheme.