In this paper, we consider distributed maximum likelihood (ML) classification of digital amplitude-phase modulated signals using multiple sensors that observe the same sequence of unknown symbol transmissions over nonidentical flat block-fading Gaussian noise channels. A variant of the expectation-maximization (EM) algorithm is employed to obtain the ML estimates of the unknown channel parameters and compute the global log-likelihood of the observations received by all the sensors in a distributed manner by means of an average consensus filter. This procedure is repeated for all candidate modulation formats in the reference library, and a classification decision, which is available at any of the sensors in the network, is declared in favor of the modulation with the highest log-likelihood score. The proposed scheme improves the classification accuracy by exploiting the signal-to-noise ratio (SNR) diversity in the network while restricting the communication to a small neighborhood of each sensor. Numerical examples show that the proposed distributed EM-based classifier can achieve the same classification performance as that of a centralized classifier, which has all the sensor measurements, for a wide range of SNR values.