We study the design of iterative receiver structures for a simple communication system, operating over an AWGN channel with unknown gain and unknown noise power. Based on a generic message passing framework, which contains Belief Propagation (BP), Variational Message Passing (VMP), and Expectation Maximization (EM) as special cases, we first rederive a non-Bayesian EM-based SNR estimator which has been proposed before in the literature. We then switch to a Bayesian model and derive a refined algorithm, which uses VMP instead of EM for estimating the channel gain and noise precision. We demonstrate via simulations that the proposed VMP-based SNR estimator outperforms the EM-based estimator in terms of a lower frame error rate, at hardly any increase of computational complexity. While we focus on coherent SNR estimation in this work, we briefly discuss a possible extension to the non-coherent case.