This paper deals with distributed coding for sensor networks. We consider energy-and-complexity-constrained sensors that communicate with a distant sink at a relatively low data rate with the help of a relay with higher-power and better-processing capabilities. We assume an orthogonal multiple access to the resources. Sources transmit independently and use simple short block codes to limit delay and complexity. The relay performs network encoding of the detected codewords and forwards either partial or complete coded information. The sink jointly decodes observations coming both from sources and relay to simultaneously estimate the information sent by the source nodes. We first analyse the cooperative scheme from an information theoretical point of view in order to determine the resource allocation strategy between the sources and the relay, which maximizes the overall network sum-rate. When a single relay is used and the source-to-relay link is not error-free, occasional detection errors introduce additional errors at the relay output, which, in turn, critically affect decoding performance and yield error-floors at the destination. In the case of a distributed coding scheme based on product codes, we obtain a closed-form asymptotically-tight approximation of the error probability under a minimum-distance decoding of the product code at the destination. The proposed analysis takes explicitly into account the error propagation at the relay output and may be extended to other forms of distributed coding schemes.