The distributed nature of wireless sensor networks illustrates well classical engineering tradeoffs: how to minimize communication (and possibly computation) cost, and thus energy dissipation, while maintaining acceptable performance levels in estimation and inference applications. We study a simple sensor network under dependent Gaussian noise and develop strategies for parameter estimation in a variety of communication scenarios. From an energy point of view, sending all data to a fusion center is the most costly, but leads to optimum performance results. Processing data at each sensor and sending parameter estimates and associated quality measures is a reasonable communication saving procedure and yet, in some cases, may lead to performance equivalent to sending all data to the fusion center. A sequential procedure is most parsimonious in terms of communication cost and especially effective in large wireless sensor networks. We explore those conditions for which little, or no loss in performance is encountered with this sequential procedure. Specifically, we provide analytical expressions for the maximum likelihood estimator under "geometric" dependent noise. We show, by means of analysis and simulations, that the performance is only marginally degraded when the noise is assumed to be independent.