This work develops an exact converging algorithm for the solution of a distributed optimization problem with partially-coupled parameters across agents in a multi-agent scenario. In this formulation, while the network performance is dependent on a collection of parameters, each individual agent may be influenced by only a subset of the parameters. Problems of this type arise in several applications, most notably in distributed control formu- lations and in power system monitoring. The resulting coupled exact diffusion strategy is shown to converge to the true optimizer at a linear rate for strongly-convex cost functions