This paper is concerned with linear state estimation in sensor networks with an event-triggered exchange of information. It is assumed that each sensor node transmits its local estimate to a fusion center whenever an appropriately chosen error norm exceeds a threshold. The fusion rule is a modified version of the Covariance Intersection Algorithm. We investigate how to incorporate the event information of not having transmitted at the fusion center such that the filter remains unbiased and consistent with regard to its error covariance. An upper bound on the error covariance matrix is derived by exploiting the structure of the posterior probability distribution. This enables us to replace the event information by the virtual transmission of consistent local estimates. Based on the consistency-preserving property of the proposed scheme, we show stability of the event-triggered state estimator in terms of a bounded mean square error.