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We consider the problem of decentralized binary detection in a sensor network where the sensors have access to side information that affects the statistics of their measurements, or reflects the quality of the available channel to a fusion center. 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 mapping from measurements to messages. We consider the case of a large number of sensors, and an asymptotic criterion involving error exponents. We study both a Neyman-Pearson and a , Bayesian formulation, characterize the optimal error exponent, and derive asymptotically optimal strategies for the case where sensor decisions are only allowed to depend on locally available information. Furthermore, we show that for the Neyman-Pearson case, global sharing of side information ("sensor cooperation") does not improve asymptotic performance, when the Type I error is constrained to be small.