This article discusses the challenges of Indigenous research in relation to open science, more particularly in relation to Sámi research in Sweden. Based on interviews with active scholars in the multidisciplinary field of Sámi studies, and on policy documents by Sámi organizations, this article points at the challenges that can be identified, and the practices and strategies adopted or suggested by researchers. Topics addressed include ownership, control, sensitivity and accessibility of data, the consequences of experienced limitations, the role of the historical context, and community-groundedness. This article has the ambition to contribute with a discussion about the tensions between standards of data management/open science and data sovereignty in Indigenous contexts. This is done by bringing in perspectives from Indigenous methodologies (the 4 R) and by contextualizing research practices and forms of data colonialism in relation to our contemporary context of surveillance culture. Research—in relation to ethics and social sustainability—is an arena where tensions between various agendas becomes obvious. This is illustrated in this article by researchers' dilemmas when working with open science and the advancement of Indigenous research. Efforts toward ethically valid and cultural-sensitive modes of data use are taking shape in Indigenous research, calling for an increased awareness about the topic. In the context of Sámi research, the role of academia in such a transformation is also essential.