This paper offers an update - from the European perspective - to the debate on property rights in personal data. It argues that recent developments in the data processing technology and practices, specifically, the AI-driven Big Data Analytics, have rendered personal data a difficult object of enforceable individual property rights. There are two main reasons for this. First, data processing resulting from a decision of one person will inevitably have spill-over effects on others, e.g. as a result of profiling, or as a result of the same piece of data relating to a group of people, e.g. genetic data. This phenomenon is also called ‘network effect’. Therefore, true individual control over personal data and also the effective enforcement of the individual property rights in personal data are difficult if not impossible to achieve. Second, creating and managing property rights that are transparent in terms of the object of property and establishing the rights-holders is also challenging. This is due to the dynamic approach to the definition of personal data adopted in Europe: the same piece of data, depending on a particular context, can be personal and non-personal, more or less likely to relate to an identifiable natural person, and with a stronger or weaker link to that person. While it is debatable if this necessarily constitutes a problem for the purposes of applying the data protection law, and if the broadest definition of personal data can achieve the goals of complete and effective protection, enforcing property rights in personal data is difficult. The difficulty lies, first, in determining at which point the level of relation to an individual is sufficient to establish property rights, and second, in tracing the presence of such a relation to a person, the ‘data owner’.