We consider a distributed detection problem where two nodes, or decision makers, observe a common source and aim to decide on one of several hypotheses. Before making their individual decisions, the nodes are allowed to communicate over rate-constrained links, through a bidirectional relay. We show that if the rate of the common relay-to-node link is greater than or equal to the rate of the individual node-to-relay links, and the individual decisions are not coupled by the cost metric, then network coding at the relay allows the overall problem to decouple into two separate two-node distributed detection problems over serial networks; and the two serial networks can be designed independently. However, if the rate of the relay-to-node link is strictly less than the node-to-relay links, no such decoupling can be assumed in general, and the overall detection network needs to be jointly designed.