In this letter, functional duality between distributed source coding (DSC) with correlated messages and broadcast channel coding (BCC) with correlated messages is considered. It is shown that under certain conditions, for a given DSC problem with correlated messages, a functional dual BCC problem with correlated messages can be obtained, and vice versa. That is, the optimal encoder-decoder mappings for one problem become the optimal decoder-encoder mappings for the dual problem. Furthermore, the correlation structure of the messages in the two dual problems and the source distortion and channel cost measure for this duality are specified.