In this paper, we investigate a distributed estimation problem for multiagent systems with state equality constraints (SEC). First, under a time-based consensus communication protocol, applying a modified projection operator and the covariance intersection fusion method, we propose a distributed Kalman filter with guaranteed consistency and satisfied SEC. Furthermore, we establish the relationship between consensus step, SEC, and estimation error covariance in dynamic and steady processes, respectively. Employing a space decomposition method, we show that the error covariance in the constraint set can be arbitrarily small by setting a sufficiently large consensus step. Besides, we propose an extended collective observability (ECO) condition based on SEC, which is milder than existing observability conditions. Under the ECO condition, through utilizing a technique of matrix approximation, we prove the boundedness of error covariance and the exponentially asymptotic unbiasedness of state estimate, respectively. Moreover, under the ECO condition for linear time-invariant systems with SEC, we provide a novel event-triggered communication protocol by employing the consistency, and give an offline design principle of triggering thresholds with guaranteed boundedness of error covariance. More importantly, we quantify and analyze the communication rate for the proposed event-triggered distributed Kalman filter, and provide optimization based methods to obtain the minimal (maximal) successive nontriggering (triggering) times. Two simulations are provided to demonstrate the developed theoretical results and the effectiveness of the filters.