Estimation based on received signal strength (RSS) is crucial in sensor networks for sensor localization, target tracking, etc. In this paper, we present a Gaussian approximation of the Chi distribution that is applicable to general RSS source localization problems in sensor networks. Using our Gaussian approximation, we provide a factorized variational Bayes (VB) approximation to the location and power posterior of multiple sources using a sensor network. When the source signal and the sensor noise have uncorrelated Gaussian distributions, we demonstrate that the envelope of the sensor output can be accurately modeled with a multiplicative Gaussian noise model. In turn, our factorized VB approximations decrease the computational complexity and provide computational robustness as the number of targets increases. Simulations are provided to demonstrate the effectiveness of the proposed approximations.