In many scenarios of interest, agents may only have access to partial information about an unknown model or target vector. Each agent may be sensing only a subset of the entries of a global target vector, and the number of these entries can be different across the agents. If each of the agents were to solve an inference task independently of the other agents, then they would not benefit from neighboring agents that may be sensing similar entries. This work develops cooperative distributed techniques that enable agents to cooperate even when their interactions are limited to exchanging estimates of select few entries. In the proposed strategies, agents are only required to share estimates of their common entries, which results in a significant reduction in communication overhead. Simulations show that the proposed approach improves both the performance of individual agents and the entire network through cooperation.