Distributed state estimation is a relevant research topic due to its application opportunities in different fields, such as multi-robot cooperation and control of large-scale networked systems. In addition, event-triggering mechanisms have been studied in recent years to reduce communication between network nodes without significantly compromising the desired behavior. In this work, we contribute a distributed algorithm to estimate the state of a stochastic system under event-triggered communication. The proposal uses consensus on the state estimates, and it takes advantage of the asymptotic form of the well-known Kalman-Bucy filter so that only state information needs to be transmitted during the online execution. We provide guarantees of boundedness of the error covariance for the state estimates under event-triggered communication. Moreover, we show that the centralized optimal solution can be recovered when the event threshold is decreased, which is an improvement with respect to existing event-triggered estimators in the stochastic context. Finally, we show via simulation that the proposal effectively reduces communication without sacrificing the quality of the estimates, and it improves performance with respect to previous approaches.