The vast amount of available medical records has the potential to improve healthcare and biomedical research. However, privacy restrictions make these data accessible for internal use only. Recent works have addressed this problem by generating synthetic data using Causal Language Modeling. Unfortunately, by taking this approach, it is often impossible to guarantee patient privacy while offering the ability to control the diversity of generations without increasing the cost of generating such data. In contrast, we present a system for generating synthetic free-text medical records using Masked Language Modeling. The system preserves critical medical information while introducing diversity in the generations and minimising re-identification risk. The system's size is about 120M parameters, minimising inference cost. The results demonstrate high-quality synthetic data with a HIPAA-compliant PHI recall rate of 96% and a re-identification risk of 3.5%. Moreover, downstream evaluations show that the generated data can effectively train a model with performance comparable to real data.