We study a multiple-terminal joint source-channel coding problem, where two remote correlated Gaussian sources are transmitted over a Gaussian multiple-access channel with two transmitters. Each transmitter observes one of the sources contaminated in Gaussian noise. The receiver wishes to reconstruct both sources. We derive necessary conditions and sufficient conditions for the receiver to be able to reconstruct the sources with given expected squared-error distortions. These conditions establish the optimality of uncoded transmission below some signal-to-noise ratio (SNR) threshold, and they also establish the high-SNR asymptotics. To achieve the latter, a coding scheme is proposed that superimposes analog uncoded transmission and digital combined source-channel Gaussian vector quantization.