Synthetic Aperture Radar (SAR) provides valuable information about the Earth's surface in all-weather and dayand-night conditions. Due to the inherent presence of speckle phenomenon, a filtering step is often required to improve the performance of downstream tasks. In this paper, we focus on dealing with the spatial correlations of speckle, which impacts negatively many of the existing speckle filters. Taking advantage of the flexibility of variational methods based on the Plug-and-Play strategy, we propose to use a Gaussian denoiser trained to restore SAR scenes corrupted by colored Gaussian noise with correlation structures typical of a range of radar sensors. Our approach improves the robustness of Plug-and-Play despeckling techniques. Experiments conducted on simulated and real polarimetric SAR images show that the proposed method removes speckle efficiently in the presence of spatial correlations without introducing artifacts, with a good level of detail preservation. Our method can be readily applied, without network retraining or fine-tuning, to filter SAR images from various sensors, acquisition modes (SAR, PolSAR, InSAR, PolInSAR), spatial resolution, and even benefit from co-registered multi-temporal stacks, when available. The code of the trained models is made freely available at https://gitlab.telecom-paris.fr/ring/mulog-drunet.