The upcoming data economy is fully supported by the fact that data are valuable. In this context, issues such as data property or data sovereignty become relevant for citizens and governments. Different regulations such as the General Data Protection Regulation in the European Union, or the Data Act Political Agreement, are in charge of preserving the rights of data owners (typically citizens) at legal level. But, in many cases, laws describe or recognize rights that cannot be provided from a technological or engineered point of view. In fact, in most digital services, users must give their data to service providers and they have no option. As, in general, service architectures consider a unique centralized data repository owned by the provider. This data transfer implies an inevitable loss of data sovereignty because there is no mechanism available for users to execute its governance. In this context, innovative service architectures are needed, in order to facilitate a distributed and effective data sovereignty to citizens. Therefore, in this paper, we present a novel Blockchain-enabled architecture for chat applications, enabling users to exercise a real data governance. The application connects users through a Blockchain network where Smart Contracts manage contacts and message exchanges in three different kinds of chats: individual public chats, group public chats, and secret chats. Data are encrypted using a combination of symmetric and asymmetric techniques and keys, so data owners can control access to their messages and information dynamically and without transferring them to any external repository. An experimental validation is provided using a pilot deployment. Results show that the architecture is scalable and provides an acceptable Quality-of-Experience to potential users.