To strike a balance between energy efficiency and data quality control, this paper proposes a Neyman-Pearson type sensor censoring scheme for distributed sparse signal recovery via compressive-sensing based on wireless sensor networks. In the proposed approach, each sensor node employs a sparse sensing vector with known support for data compression, meanwhile enabling making local inference about the unknown support of the sparse signal vector of interest. This naturally leads to a ternary censoring protocol, whereby each sensor (i) directly transmits the real-valued compressed data if the sensing vector support is detected to be overlapped with the signal support, (ii) sends a one-bit hard decision if empty support overlap is inferred, (iii) keeps silent if the measurement is judged to be uninformative. Our design then aims at minimizing the error probability that empty support overlap is decided but otherwise is true, subject to the constraints on a tolerable false-alarm probability that non-empty support overlap is decided but otherwise is true, and a target censoring rate. We derive a closed-form formula of the optimal censoring rule; a low complexity implementation using bi-section search is also developed. Computer simulations are used to illustrate the performance of the proposed scheme.