The sharing of biomedical data is essential for accelerating healthcare research, fostering medical innovation, and improving patient outcomes. Such data encompasses a wide range of sensitive information, including electronic health records, genomic sequences, and clinical trial results. Despite its value, biomedical data sharing poses significant privacy risks, such as patient re-identification, unauthorized access, and regulatory non-compliance. These concerns necessitate advanced techniques that balance the need for data utility with stringent privacy protection. Machine learning (ML) has emerged as a powerful tool to facilitate privacy-preserving biomedical data sharing. This manuscript presents a comprehensive review of state-of-the-art ML-based privacy preservation methods, including differential privacy, federated learning, homomorphic encryption, secure multi-party computation, and synthetic data generation through generative models. Each technique offers unique mechanisms to protect sensitive information while enabling collaborative analysis and predictive modeling. These methods have been applied practically across various biomedical domains, including collaborative disease risk prediction and genomic research, clinical trial data analysis, remote patient monitoring, and public health surveillance. Additionally, we evaluate relevant privacy and utility metrics that assess the effectiveness of privacy guarantees and the impact on model performance. The review further examines limitations and challenges—including computational overhead, data heterogeneity, privacy-utility trade-offs, and ethical considerations—that must be addressed to ensure robust and scalable solutions. Looking forward, the manuscript highlights promising future directions, such as hybrid privacy frameworks, enhanced synthetic data generation, real-time privacy-preserving analytics, standardization of evaluation protocols, and interdisciplinary policy development. By integrating these advancements, biomedical research can achieve safer and more effective data sharing, ultimately fostering innovation while respecting patient confidentiality and trust.