This paper develops the sufficiency principle suitable for data reduction in decentralized inference systems. Both parallel and tandem networks are studied and we focus on the cases where observations at decentralized nodes are conditionally dependent. For a parallel network, through the introduction of a hidden variable that induces conditional independence among the observations, the locally sufficient statistics, defined with respect to the hidden variable, are shown to be globally sufficient for the parameter of inference interest. For a tandem network, the notion of conditional sufficiency is introduced and the related theories and tools are developed. Finally, connections between the sufficiency principle and some distributed source coding problems are explored.