The introduction of data analytics into medicine has changed the nature of patient treatment. In this, patients are asked to disclose personal information such as genetic markers, lifestyle habits, and clinical history. This data is then used by statistical models to predict personalized treatments. However, due to privacy concerns, patients often desire to withhold sensitive information. This self-censorship can impede proper diagnosis and treatment, which may lead to serious health complications and even death over time. In this paper, we present privacy distillation, a mechanism which allows patients to control the type and amount of information they wish to disclose to the healthcare providers for use in statistical models. Meanwhile, it retains the accuracy of models that have access to all patient data under a sufficient but not full set of privacy-relevant information. We validate privacy distillation using a corpus of patients prescribed to warfarin for a personalized dosage. We use a deep neural network to implement privacy distillation for training and making dose predictions. We find that privacy distillation with sufficient privacy-relevant information i) retains accuracy almost as good as having all patient data (only 3% worse), and ii) is effective at preventing errors that introduce health-related risks (only 3.9% worse under- or over-prescriptions).