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In this paper, we introduce a framework for Joint Source-Channel Coding of distributed Gaussian sources over a multiple access AWGN channel. Although there are prior works that have studied this, they either strongly rely on intuition to design encoders and decoder or require the knowledge of the complete joint distribution of all the distributed sources. Our system overcomes this. We model our system as a Variational Autoencoder and leverage insight provided by this connection to propose a crucial regularization mechanism for learning. This allows us to beat the state of the art by improving the signal reconstruction quality by almost 1dB for certain configurations. The end-to-end learned system is also found to be robust to channel condition variations of ±5dB and shows a drop in signal reconstruction quality by at most 1dB. Finally, we propose a novel lower bound on the optimal distortion in signal reconstruction and empirically showcase the tightness of the bound in comparison with the existing bound.