A distributed data selection technique for fusion center (FC)-based estimation with a wireless sensor network (WSN) is presented. The data selection is envisioned for a large WSN in which only a subset of measurements need be transmitted to the FC thereby saving on transmission power. Furthermore, quantization of the selected measurements leading to bandwidth savings is also addressed. A novel data selection method using measurement censoring is followed by maximum a posteriori estimation that optimally fuses information from the censored-data model. Censoring and estimation algorithms that are amenable to implementation with WSNs are developed. Bayesian Cramér-Rao bound analysis and numerical simulations show that the proposed censoring-based estimator and quantized-censored estimator have competitive (or even superior) mean-square error performance when compared to data selection alternatives under a range of sensing conditions.