Diffusion-based synthetic data generation offers a promising route for sharing medical imaging data without releasing sensitive patient records. However, generative models face a fundamental tension between privacy and fairness: they may memorize rare training samples, leading to privacy risks, or fail to reproduce underrepresented features, resulting in unfair synthetic distributions. While prior work has largely focused on either memorization or fairness in isolation, their interaction remains insufficiently understood. In this work, we introduce a data-interventional framework to systematically analyze privacy and fairness in diffusion models. We discuss synthetic anatomical fingerprints (SAFs), rare and manually injected image features, as controlled probes to study whether models generalize sensitive attributes across identities, memorize training samples, or suppress rare signals entirely. Across multiple conditioning modalities, we observe a consistent behavior: models either forget these fingerprints or memorize the entire image in which they appear, but do not generalize them to novel images. To support large-scale auditing where explicit sample extraction is infeasible, we further introduce the indicator metric t', which estimates a model's susceptibility to memorization by exploiting the internal structure of the diffusion process. By comparing conditioning signals of varying surprisal, we reveal a clear relationship between conditioning rarity and memorization behavior. Highly surprising conditioning signals act as retrieval keys that amplify memorization, whereas low-surprisal conditioning signals systematically suppress rare features, even when these appear repeatedly in the training data. Our findings provide actionable insights and concrete mitigation strategies for safe and fair synthetic medical data sharing. Code is available at https://github.com/MischaD/Privacy.

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