In this paper, we consider a periodic estimation problem in sensor networks with a shared communication channel. The transmission constraint is inevitable in a single-channel-based sensor network if the sensors are heterogeneous or deployed far away from each other. A novel stochastic competitive transmission strategy is presented to deal with the transmission constraint, such that the sensors communicate with the fusion center (FC) in a strict asynchronous manner. A periodic mixed storage strategy combing the zero-input and the hold-input mechanisms is presented to describe periodic updating of the stored information in the sensors' buffers. A recursive Kalman filtering algorithm is derived for the FC to periodically generate estimates of state variables describing an object by using a linear continuous-time stochastic model. Two simulation examples are presented to show the effectiveness of the proposed results.