In this paper, the privacy problem of a parallel distributed detection system vulnerable to an eavesdropper is proposed and studied in the Bayesian formulation. The privacy risk is evaluated by the detection cost of the eavesdropper which is assumed to be informed and greedy. It is shown that the optimal detection strategy of the sensor whose decision is eavesdropped on is a likelihood-ratio test. This fundamental insight allows for the optimization to reuse known algorithms extended to incorporate the privacy constraint. The trade-off between the detection performance and privacy risk is illustrated in a numerical example. The incorporation of physical layer privacy in the system design will lead to trustworthy sensor networks in future.