This paper examines distributed event-triggered estimation over wireless sensor networks. In such systems an efficient utilization of the wireless communications must be performed since energy consumption and communication bandwidth are limit resources. We pose a global event-triggered communication policy for state estimation that minimizes a weighted function of the network energy consumption and the number of transmissions subject to constraints on the estimator performance. The global communication policy determines when sensors transmit measurements to the central estimator using a sensor-to-estimator communication channel as well as when sensors received other sensors' measurements (which have been transmitted to the central estimator) using a estimator-to-sensor communication channel. A distributed 1-step greedy heuristic is introduced for the proposed global minimization problem such that sensors determine their respective communication policies using only the local information available at each sensor. Simulation results demonstrate that the number of sensor transmissions can be reduced at a potential increase in network energy consumption (number of sensor transmission and receptions) with the added benefit of reducing network congestion.