Denoising diffusion models (DDMs) are state-of-the-art for numerous tasks pertaining to image generation and inverse problems. Yet many aspects of the training and sampling pipeline remain poorly understood. For instance, the necessity of noise conditioning has been particularly elusive, forcing practitioners to incorporate unnatural noise embeddings into neural network architectures and use ad hoc noise schedules during sampling. Motivated to remove these inconsistencies, we provide a complete theory for a \emph{blind denoising diffusion models} (BDDMs): a variant of DDMs where the noise amplitude is not passed into the neural network during training nor sampling. We justify the correctness of BDDMs as a sampling algorithm under the sole assumption of low intrinsic dimensionality of the underlying data distribution relative to the ambient dimension. This assumption arises through the introduction of the Bayesian problem of estimating noise levels through a single noisy sample, which might be of independent interest. We compare the performance of BDDMs to standard DDMs in a variety of settings, showcasing the benefits of an adaptive scheme which is rigorously justified by our theoretical analysis.