In this paper, we consider the problem of source coding for a wireless acoustic sensor network where each node in the network makes its own noisy measurement of the sound field, and communicates with other nodes in the network by sending and receiving encoded versions of the measurements. To make use of the correlation between the sources available at the nodes, we consider the possibility of combining the measurement and the received messages into one single message at each node instead of forwarding the received messages and separate encoding of the measurement. Moreover, to exploit the correlation between the messages received by a node and the node's measurement of the source, we propose to use the measurement as side information and thereby form a distributed source coding (DSC) problem. Assuming that the sources are Gaussian, we then derive the rate-distortion function (RDF) for the resulting remote DSC problem under covariance matrix distortion constraints. We further show that for this problem, the Gaussian source is the worst to code. Thus, the Gaussian RDF provides an upper bound to other sources such as audio signals. We then turn our attention to audio signals. We consider an acoustical model based on the room impulse response (RIR) and provide simulation results for the rate-distortion performance in a practical setup where a set of microphones record the sound in a standard listening room. Since our reconstruction scheme and distortion measure are defined over the direct sound source, coding and dereverberation are performed in a joint manner.