This paper explores integrated source-channel decoding, driven by wireless sensor network applications where correlated information acquired by the network is gathered at a destination node. The collection of coded measurements sent to the destination, called a source-channel product codeword, has redundancy due to both correlation of the measurements and the channel code used for each measurement. At the destination, source-channel (SC) decoding of this code combines decoding using (i) the deterministic structure of the channel-coded individual measurements and (ii) the probabilistic structure of a prior model, called the global model, that describes the correlation structure of the SC product codewords. We demonstrate the utility of SC decoding via MAP SC decoding experiments using a (7,4,3) Hamming code and a Gaussian global model. We also show that SC decoding can exploit even the simplest possible code, a single-parity check code, using a MAP SC decoder that integrates the parity check constraint and global model. We describe the design of a low-complexity message-passing decoder and show it can improve performance in the poor-quality channels often found in multi-hop wireless data-gathering sensor networks.