In this paper, we design and analyze distributed vector quantization (VQ) for compressed measurements of correlated sparse sources over noisy channels. Inspired by the framework of compressed sensing (CS) for acquiring compressed measurements of the sparse sources, we develop optimized quantization schemes that enable distributed encoding and transmission of CS measurements over noisy channels followed by joint decoding at a decoder. The optimality is addressed with respect to minimizing the sum of mean-square error (MSE) distortions between the sparse sources and their reconstruction vectors at the decoder. We propose a VQ encoder-decoder design via an iterative algorithm, and derive a lower-bound on the end-to-end MSE of the studied distributed system. Through several simulation studies, we evaluate the performance of the proposed distributed scheme.