This paper deals with demodulation of space-timetransmissions from a multi-antenna access point to a network of spatially distributed wireless sensors. Distributed demodulation algorithms are developed by achieving network-wide consensus on the average of (cross-) covariances of locally available per sensor received data vectors with the channel matrix, which constitute sufficient statistics for maximum likelihood demodulation. By reaching consensus on such average terms, each sensor can attain demodulation performance as if all the information was available at a centralized unit. Different from existing distributed hypotheses testing schemes whose complexity grows exponentially with the problem dimension, the novel consensusbased demodulator incurs quadratic complexity. Inter-sensor link imperfections due to additive noise and random link failures are also accounted for. Consensus in these cases is achieved in the mean sense with bounded variance, and in the mean-square error sense, respectively. Simulated tests verify the analytical claims. Interestingly, only a few consensus iterations suffice for the novel distributed demodulator to approach the performance of its centralized benchmark.