SAR despeckling is a problem of paramount importance in remote sensing, since it represents the first step of many scene analysis algorithms. Recently, deep learning techniques have outperformed classical model-based despeckling algorithms. However, such methods require clean ground truth images for training, thus resorting to synthetically speckled optical images since clean SAR images cannot be acquired. In this paper, inspired by recent works on blind-spot denoising networks, we propose a self-supervised Bayesian despeckling method. The proposed method is trained employing only noisy images and can therefore learn features of real SAR images rather than synthetic data. We show that the performance of the proposed network is very close to the supervised training approach on synthetic data and competitive on real data.