The Data Act proposal of February 2022 constitutes a central element of a broader and ambitious initiative of the European Commission (EC) to regulate the data economy through the erection of a new general regulatory framework for data and digital markets. The resulting framework may be represented as a model of governance between a pure market-driven model and a fully regulated approach, thereby combining elements that traditionally belong to private law (e.g., property rights, contracts) and public law (e.g., regulatory authorities, limitation of contractual freedom). This article discusses the role of (intellectual) property rights as well as of other forms of rights allocation in data legislation with particular attention to the Data Act proposal. We argue that the proposed Data Act has the potential to play a key role in the way in which data, especially privately held data, may be accessed, used, and shared. Nevertheless, it is only by looking at the whole body of data (and data related) legislation that the broader plan for a data economy can be grasped in its entirety. Additionally, the Data Act proposal may also arguably reveal the elements for a transition from a property-based to a governance-based paradigm in the EU data strategy. Whereas elements of data governance abound, the stickiness of property rights and rhetoric seem however hard to overcome. The resulting regulatory framework, at least for now, is therefore an interesting but not always perfectly coordinated mix of both. Finally, this article suggests that the Data Act Proposal may have missed the chance to properly address the issue of data holders’ power and related information asymmetries, as well as the need for coordination mechanisms.