This paper studies the nonlinear distributed filtering problem for a class of discrete state-saturated systems with censoring measurements and multiplicative noises under the dynamic event-triggered communication mechanism (DETCM). The censoring measurements, which are characterized by the Tobit type I model, unavoidably arise in the measurement signals resulted from the sudden changes of circumstances or the repairs of components. Meanwhile, for the sake of reducing the communication cost and improving the data transmission efficiency, the DETCM is used to determine when the sensor has an opportunity to transmit the information to its adjacent nodes according to the fixed communication topology. The main purpose of the paper is to design a novel distributed filter under the DETCM for the underlying sensor networks, where a particular upper bound of the filtering error covariance is obtained by means of the mathematical induction approach and the matrix theory. Then, the proposed upper bound can be minimized via designing the suitable filter gain. Further, the boundedness of filtering error dynamics is proved based on some pre-set conditions. Finally, illustrative simulation experiments are provided to illustrate the effectiveness of the presented distributed filtering algorithm.