We consider the distributed hypothesis testing problem, and specifically, a scenario where the hypotheses regard the joint statistics of two sequences, one available to the decision function directly (``side information'') while the other is conveyed through a limited-rate link. We are interested in the optimal false-alarm and misdetection exponents. We show that if 
the hypotheses induce the same marginals with respect to the sequence to be compressed, the problem is equivalent to one of channel detection, in the spirit of previous works that show equivalence between the Slepian-Wolf problem and channel coding. For the equivalent channel problem, we present the exponents of optimal decoding when the input is drawn from a random or an expurgated ensemble. We discuss a possible improvement of the exponents by using hierarchical ensembles, and suboptimal strategies for bounding the corresponding exponents.