Digitization triggered a steep drop in the cost of information. The resulting data glut created a bottleneck because human cognitive capacity is unable to cope with large amounts of information. Artificial intelligence and machine learning (AI/ML) triggered a similar drop in the cost of machine-based decision-making and helps in overcoming this bottleneck. Substantial change in the relative price of resources puts pressure on ownership and access rights to these resources. This explains pressure on access rights to data. ML thrives on access to big and varied datasets. We discuss the implications of access regimes for the development of AI in its current form of ML. The economic characteristics of data (non-rivalry, economies of scale and scope) favour data aggregation in big datasets. Non-rivalry implies the need for exclusive rights in order to incentivise data production when it is costly. The balance between access and exclusion is at the centre of the debate on data regimes. We explore the economic implications of several modalities for access to data, ranging from exclusive monopolistic control to monopolistic competition and free access. Regulatory intervention may push the market beyond voluntary exchanges, either towards more openness or reduced access. This may generate private costs for firms and individuals. Society can choose to do so if the social benefits of this intervention outweigh the private costs. We briefly discuss the main EU legal instruments that are relevant for data access and ownership, including the General Data Protection Regulation (GDPR) that defines the rights of data subjects with respect to their personal data and the Database Directive (DBD) that grants ownership rights to database producers. These two instruments leave a wide legal no-man's land where data access is ruled by bilateral contracts and Technical Protection Measures that give exclusive control to de facto data holders, and by market forces that drive access, trade and pricing of data. The absence of exclusive rights might facilitate data sharing and access or it may result in a segmented data landscape where data aggregation for ML purposes is hard to achieve. It is unclear if incompletely specified ownership and access rights maximize the welfare of society and facilitate the development of AI/ML.