This paper addresses the problem of correct recovery of multiple sparse correlated signals using distributed thresholding. We consider the scenario where multiple sensors capture the same event, but observe different signals that are correlated by local transforms of their sparse components. In this context, the signals do not necessarily have the same sparse support, but instead the support of one signal is built on local transformations of the atoms in the sparse support of another signal. We establish the sufficient condition for the correct recovery of such correlated signals using independent thresholding of the multiple signals. The validity of the derived recovery condition is confirmed by experimental results in noiseless and noisy scenarios.