  This paper presents a communication efficient distributed algorithm,
$\mathcal{CIRFE}$ of the \emph{consensus}+\emph{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~(\emph{innovations}) and the
parameter estimates from agents~(\emph{consensus}) in its communication
neighborhood given by a time-varying possibly sparse graph. Under minimal
conditions on the inter-agent 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 $\mathcal{CIRFE}$ 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 $\mathcal{CIRFE}$.
