The paper presents a communication efficient distributed algorithm, CIRFE of the consensus+innovations type, to estimate a high-dimensional parameter in a multi-agent network, in which each agent is interested in reconstructing only a few components of the parameter. This problem arises, for example, when monitoring the high-dimensional distributed state of a large-scale infrastructure with a network of limited capability sensors and where each sensor is tasked with estimating some local components of the state. At each observation sampling epoch, each agent updates its local estimate of the parameter components in its interest set by simultaneously processing the latest locally sensed information (innovations) and the parameter estimates from agents (consensus ) in its communication neighborhood given by a time-varying possibly sparse graph. Under minimal conditions, on the interagent communication network and the sensing models, almost sure convergence of the estimate sequence at each agent to the components of the true parameter in its interest set is established. Furthermore, the paper establishes the performance of CIRFE in terms of asymptotic covariance of the estimate sequences and specifically characterizes the dependencies of the component wise asymptotic covariance in terms of the number of agents tasked with estimating it. Finally, simulation experiments demonstrate the efficacy of CIRFE.