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Diffusion models (DMs) are a powerful frame- work for image generation and restoration. How- ever, existing DMs are primarily trained in a su- pervised manner by using a large corpus of clean images. This poses fundamental challenges in many real-world scenarios, where acquiring noise- free data is hard or infeasible. While some meth- ods are capable of training DMs using noisy data, they are effective only when the amount of noise is very mild or when additional noise-free data is available. In addition, existing methods for training DMs from incomplete measurements re- quire access to multiple complementary acquisi- tion processes, a significant practical limitation. Here we introduce the first approach for learning DMs for image restoration using only noisy mea- surement data from a single operator. First, we show that DMs, and more broadly minimum mean squared error denoisers, exhibit a weak form of scale equivariance linking rescaling in signal am- plitude to changes in noise intensity. We then leverage this theoretical insight to develop a de- noising score-matching strategy that generalizes robustly to noise levels below the training data, thereby enabling the learning of DMs from noisy measurements. For problems involving measure- ments both noisy and incomplete, we integrate our method with equivariant imaging, a complemen- tary self-supervised learning framework that ex- ploits the inherent invariants of imaging problems. This allows training DMs for image restoration from single-operator noisy measurements. We validate the effectiveness of our approach through extensive experiments on image denoising, demo- saicing, inpainting, and MRI reconstruction along with comparisons with the state of the art.