  The problem of distributed binary hypothesis testing in the Gray-Wyner
network with side information is studied in this paper. An observer has access
to a discrete memoryless and stationary source and describes its observation to
two detectors via one common and two private channels. The channels are
considered error-free but rate-limited. Each detector also has access to its
own discrete memoryless and stationary source, i.e., the side information. The
goal is to perform two distinct binary hypothesis testings on the joint
distribution of observations at detectors. Additionally, the observer aims to
keep a correlated latent source private against the detectors. Equivocation is
used as the measure of the privacy preserved for the latent source. An
achievable inner bound is derived for the general case by introducing a
non-asymptotic account of the output statistics of the random binning.
