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We consider a decentralized estimation network subject to communication constraints such that nearby platforms can communicate with each other through low capacity links rendering an undirected graph. After transmitting symbols based on its measurement, each node outputs an estimate for the random variable it is associated with as a function of both the measurement and incoming messages from neighbors. We are concerned with the underlying design problem and handle it through a Bayesian risk that penalizes the cost of communications as well as estimation errors, and constraining the feasible set of communication and estimation rules local to each node by the undirected communication graph. We adopt an iterative solution previously proposed for decentralized detection networks which can be carried out in a message passing fashion under certain conditions. For the estimation case, the integral operators involved do not yield closed form solutions in general so we utilize Monte Carlo methods. We achieve an iterative algorithm which yields an approximation to an optimal decentralized estimation strategy in a person by person sense subject to such constraints. In an example, we present a quantification of the trade-off between the estimation accuracy and cost of communications using the proposed algorithm.