We consider the problem of Bayesian decentralized binary detection in a sensor network in which the sensors have access to some side information that affects the statistics of the measurements they make. Sensors can decide whether or not to make a measurement and transmit a message to the fusion center ("censoring"), and also have a choice of the transmission function from measurements to messages. We consider the case of a large number of sensors, characterize the optimal error exponent, and derive asymptotically optimal strategies. We show that the optimal strategy consists of dividing the sensors into two groups, with sensors in each group using the same policy.