This paper considers the joint compression of a pair of correlated sources, where the encoder is allowed to access only one of the sources. The objective is to recover both sources under separate distortion constraints for each source while minimizing the rate. This problem generalizes the indirect lossy source coding problem by also requiring the recovery of the observed source. In this paper, we aim to study the nonasymptotic and second-order asymptotic properties of this problem. Specifically, we begin by deriving nonasymptotic achievability and converse bounds valid for general sources and distortion measures. The source dispersion (Gaussian approximation) is then determined through asymptotic analysis of the nonasymptotic bounds. We further examine the case of erased fair coin flips (EFCF) and provide its specific nonasymptotic achievability and converse bounds. Numerical results under the EFCF case demonstrate that our second-order asymptotic approximation closely approximates the optimum rate at appropriately large blocklengths.