Artificial intelligence (AI) in healthcare promises to make healthcare safer, more accurate, and more cost-effective. Public and private actors have been investing significant amounts of resources into the field. However, to benefit from data-intensive medicine, particularly from AI technologies, one must first and foremost have access to data. It has been previously argued that the conventionally used “consent or anonymize approach” undermines data-intensive medicine, and worse, may ultimately harm patients. Yet, this is still a dominant approach in European countries and framed as an either-or choice. In this paper, we contrast the different data governance approaches in the EU and their advantages and disadvantages in the context of healthcare AI. We detail the ethical trade-offs inherent to data-intensive medicine, particularly the balancing of data privacy and data access, and the subsequent prioritization between AI and other effective health interventions. If countries wish to allocate resources to AI, they also need to make corresponding efforts to improve (secure) data access. We conclude that it is unethical to invest significant amounts of public funds into AI development whilst at the same time limiting data access through strict privacy measures, as this constitutes a waste of public resources. The “AI revolution” in healthcare can only realise its full potential if a fair, inclusive engagement process spells out the values underlying (trans) national data governance policies and their impact on AI development, and priorities are set accordingly.