In this paper, the privacy leakage problem in an eavesdropped parallel distributed binary hypothesis test network is considered. A novel Neyman–Pearson test-operational privacy leakage measure is proposed and a privacy-constrained distributed Neyman–Pearson test problem is formulated. Such privacy-constrained distributed Neyman–Pearson test network is designed to optimize the Neyman–Pearson test performance and meanwhile to satisfy a desired suppression constraint on the privacy leakage. This study characterizes the privacy-constrained distributed Neyman–Pearson test network design and particularly identifies the sufficiency of deterministic likelihood-ratio test for optimality. These results help to simplify the optimal design problem of a privacy-constrained distributed Neyman–Pearson test network. Numerical results are presented to show the trade-off between the test performance and privacy leakage in privacy-constrained distributed Neyman–Pearson test networks.