Collection of health data has increasingly become central within healthcare and public health, and various actors such as clinicians, researchers, policymakers and patients are increasingly voicing claims to being able to responsibly control data. In this context, the concept of health data sovereignty (HDS) has had highs and lows in bioethics and policy discussions over the years without being either fully discarded or established. We argue that while there is a persistent discussion on defining the underlying concepts of “health” and “data”, the real hurdle in the widescale adoption of the concept of HDS in research and policy lies in substantial disagreements on the scope and implications of the concept of sovereignty. We argue that before the concept of HDS can have a meaningful impact, we need to clarify five issues: (1) lack of consensus over who should claim sovereignty, (2) the public goods problem of sovereignty, (3) reliably maintaining sovereignty, (4) the need to protect the reasonable interests of all stakeholders, and (5) the links between solidarity and sovereignty. In mapping these conceptual challenges and the tensions between them, we hope to ignite new policy efforts and community-level grassroots initiatives towards HDS based on ethical deliberations.