  This paper proposes $\mathbf{C}$ommunication efficient $\mathbf{RE}$cursive
$\mathbf{D}$istributed estimati$\mathbf{O}$n algorithm, $\mathcal{CREDO}$, for
networked multi-worker setups without a central master node. $\mathcal{CREDO}$
is designed for scenarios in which the worker nodes aim to collaboratively
estimate a vector parameter of interest using distributed online time-series
data at the individual worker nodes. The individual worker nodes iteratively
update their estimate of the parameter by assimilating latest locally sensed
information and estimates from neighboring worker nodes exchanged over a
(possibly sparse) time-varying communication graph. The underlying inter-worker
communication protocol is adaptive, making communications increasingly
(probabilistically) sparse as time progresses. Under minimal conditions on the
inter-worker information exchange network and the sensing models, almost sure
convergence of the estimate sequences at the worker nodes to the true parameter
is established. Further, the paper characterizes the performance of
$\mathcal{CREDO}$ in terms of asymptotic covariance of the estimate sequences
and specifically establishes the achievability of optimal asymptotic
covariance. The analysis reveals an interesting interplay between the
algorithm's communication cost~$\mathcal{C}_{t}$ (over $t$ time-steps) and the
asymptotic covariance. Most notably, it is shown that $\mathcal{CREDO}$ may be
designed to achieve a $\Theta\left(\mathcal{C}_{t}^{-2+\zeta}\right)$ decay of
the mean square error~($\zeta>0$, arbitrarily small) at each worker node, which
significantly improves over the existing $
\Theta\left(\mathcal{C}_{t}^{-1}\right)$ rates. Simulation examples on both
synthetic and real data sets demonstrate $\mathcal{CREDO}$'s communication
efficiency.
