Consider the problem where a statistician in a twonode system receives rate-limited information from a transmitter about marginal observations of a memoryless process generated from two possible distributions. Using its own observations, this receiver is required to first identify the legitimacy of its sender by declaring the joint distribution of the process, and then depending on such authentication it generates the adequate reconstruction of the observations satisfying an average perletter distortion. The performance of this setup is investigated through the corresponding rate-error-distortion region describing the trade-off between: the communication rate, the error exponent induced by the detection and the distortion incurred by the source reconstruction. In the special case of testing against independence, where the alternative hypothesis implies that the sources are independent, the optimal rate-error-distortion region is characterized. An application example to binary symmetric sources is given subsequently and the explicit expression for the rate-error-distortion region is provided as well. The case of “general hypotheses” is also investigated. A new achievable rate-error-distortion region is derived based on the use of nonasymptotic binning, improving the quality of communicated descriptions. Further improvement of performance in the general case is shown to be possible when the requirement of source reconstruction is relaxed, which stands in contrast to the case of general hypotheses.