Self-supervised monocular depth estimation methods aim to be used in critical applications such as autonomous vehicles for environment analysis. To circumvent the potential imperfections of these approaches, a quantification of the prediction confidence is crucial to guide decision-making systems that rely on depth estimation. In this paper, we propose MonoProb, a new unsupervised monocular depth estimation method that returns an interpretable uncertainty, which means that the uncertainty reflects the expected error of the network in its depth predictions. We rethink the stereo or the structure-from-motion paradigms used to train unsupervised monocular depth models as a probabilistic problem. Within a single forward pass inference, this model provides a depth prediction and a measure of its confidence, without increasing the inference time. We then improve the performance on depth and uncertainty with a novel self-distillation loss for which a student is supervised by a pseudo ground truth that is a probability distribution on depth output by a teacher. To quantify the performance of our models we design new metrics that, unlike traditional ones, measure the absolute performance of uncertainty predictions. Our experiments highlight enhancements achieved by our method on standard depth and uncertainty metrics as well as on our tailored metrics. https://github.com/CEA-LIST/MonoProb