The article examines four models of data governance emerging in the current platform society. While major attention is currently given to the dominant model of corporate platforms collecting and economically exploiting massive amounts of personal data, other actors, such as small businesses, public bodies and civic society, take also part in data governance. The article sheds light on four models emerging from the practices of these actors: data sharing pools, data cooperatives, public data trusts and personal data sovereignty. We propose a social science-informed conceptualisation of data governance. Drawing from the notion of data infrastructure we identify the models as a function of the stakeholders’ roles, their interrelationships, articulations of value, and governance principles. Addressing the politics of data, we considered the actors’ competitive struggles for governing data. This conceptualisation brings to the forefront the power relations and multifaceted economic and social interactions within data governance models emerging in an environment mainly dominated by corporate actors. These models highlight that civic society and public bodies are key actors for democratising data governance and redistributing value produced through data. Through the discussion of the models, their underpinning principles and limitations, the article wishes to inform future investigations of socio-technical imaginaries for the governance of data, particularly now that the policy debate around data governance is very active in Europe.