Synthetic aperture radar (SAR) images are inherently affected by multiplicative speckle noise, which will degrade the human interpretation and computer-aided scene analysis. In this paper, we propose a novel Bayesian multiscale method for SAR image despeckling in the non-homomorphic framework. To address the multiplicative nature, we first make the speckle contribution additive by a linear decomposition. Then, in the stationary wavelet transform domain, a two-sided generalized Gamma distribution (G$Gamma$D) is introduced as a prior to capture the heavy-tailed nature of wavelet coefficients of the noise-free reflectivity. By exploiting this prior together with a Gaussian likelihood, an analytical wavelet shrinkage function is derived based on maximum a posteriori criteria, which further adopts heterogeneity-adaptive thresholding technique to achieve better estimates of noise-free wavelet coefficients. Moreover, a pilot-signal-assisted strategy is proposed to estimate the parameters of two-sided G $Gamma$D with the estimator based on second-kind cumulants. Finally, experimental results, carried out on the synthetic and actual SAR images, are given to demonstrate the validity of the proposed despeckling method.