The performance of a parallel distributed detection system is investigated as the number of sensors tends to infinity. It is assumed that the i.i.d. sensor data are quantized locally into Mary messages and transmitted to the fusion center for binary hypothesis testing. The boundedness of the second moment of the postquantization log-likelihood ratio is examined in relation to the asymptotic error exponent. It is found that when that second moment is unbounded, the Neyman-Pearson error exponent can become a function of the test level, whereas the Bayes error exponent remains, as previously conjectured by Tsitsiklis, unaffected. Large deviations techniques are also employed to show that in Bayes testing, the equivalence of absolutely optimal and best identical-quantizer systems is not limited to error exponents, but extends to the actual Bayes error probabilities up to a multiplicative constant.