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Event cameras are emerging vision sensors whose noise is challenging to characterize. Existing denoising methods for event cameras are often designed in isolation and thus consider other tasks, such as motion estimation, separately (i.e., sequentially after denoising). However, motion is an intrinsic part of event data, since scene edges cannot be sensed without motion. We propose, to the best of our knowledge, the first method that simultaneously estimates motion in its various forms (e.g., ego-motion, optical flow) and noise. The method is flexible, as it allows replacing the one-step motion estimation of the widely-used Contrast Maximization framework with any other motion estimator, such as deep neural networks. The experiments show that the proposed method achieves state-of-the-art results on the E-MLB denoising benchmark and competitive results on the DND21 benchmark, while demonstrating effectiveness across motion estimation and intensity reconstruction tasks. Our approach advances event-data denoising theory and expands practical denoising use-cases via open-source code. Project page: https://github.com/tub-rip/ESMD