We investigate channel code rates for communication of finite-dimensional analog sources over a multiple-antenna multiple access channel (MAC) so that the average end-to-end distortions are minimized. Our analysis uses the high-resolution quantization theory for the sources and the high-SNR diversity-multiplexing tradeoff for the MAC. We prove that carefully balanced channel coding rates, usually far from the boundary of the MAC capacity region, are necessary to achieve the optimal distortion exponent in a separated architecture. In particular, for the case of source vectors of equal dimension, we show that the channel interference from multiple users becomes crucial in characterizing the optimal channel coding rates if individual minimization of distortion for each user leads to a heavily loaded regime for the MAC.