This paper considers a network of sensors that collectively sense a number of unknown parameters. Each sensor can possibly sense only a subset of the parameters, gather data only about these parameters, and has access to only the statistical model of the data that it collects locally. The objective is that each sensor forms optimal estimates for its designated parameters (i.e., the parameters that it can sense). The paper proposes an estimation cost function that strikes a balance between (i) the sensors being autonomous in forming local estimates based on their locally available data and statistical models, and (ii) enforcing consistency among the local estimates formed for the parameters that are sensed by multiple sensors. Exact optimal estimators are characterized, and it is shown that the optimal estimators can be implemented in a distributed way, through a single-shot exchange of local decisions (SELD). Specifically, the distributed implementation consists of forming local estimates and exchanging certain sufficient statistics values in a single round of communication exchange among some of the sensors. Furthermore, the optimal performance under the proposed cost function is also compared analytically against the performance of the widely-used mean squared error estimator.