In this paper, we consider the problem of distributed sequential estimation in a network whose communication channels are affected by additive Gaussian noise. We propose a method that is based on cooperation among neighboring agents and that allows every agent to reach the belief that is the optimal Bayesian solution. This solution is the posterior distribution of the unknowns that is held by a fictitious fusion center. The agents, however, do not implement the Bayes' rule. Compared with the standard average consensus algorithm, the proposed method is stable in the sense that the effects of the noise do not accumulate with time and a random walk behavior is avoided. We show that with the proposed method every agent's belief converges to the belief of a fictitious fusion center, if the variance of the communication noise is bounded. We provide computer simulations that compare the proposed method with a method which works well in the noise-free case.