We study a Bayesian decentralized binary hypothesis testing problem, in which N sensors make observations related to a two-valued hypothesis, and send messages based on the observations to a fusion center, with the objective of enabling the fusion center to accurately reconstruct the realized hypothesis. We assume that the observations at the sensors are independent and identically distributed conditioned on the hypothesis, and that the sensors transmit their messages over independent, parallel channels to the fusion center. We also make the natural assumption that the sensors have identical, finite message sets at their disposal, that each message has some cost associated with it, and that there is a constraint on the average cost incurred by a sensor. We study the large N asymptote, and therefore are interested in schemes that optimize the error exponent at the fusion center. Our main contributions are 1) proving that the problem of finding the optimal schemes is a finite dimensional optimization problem, 2) a description of the structure of optimal rules: we prove that optimal sensor rules are randomized likelihood ratio quantizers (LRQs), with randomization being over at most two deterministic LRQs. We further show that under some conditions, randomization is not required.