Personalized healthcare is nowadays driven by the increasing volumes of patient data, observed and produced continuously thanks to medical devices, mobile sensors, patient-reported outcomes, among other data sources. This data is made available as streams, due to their dynamic nature, which represents an important challenge for processing, querying and interpreting the incoming information. In addition, the sensitive nature of healthcare data poses significant restrictions regarding privacy, which has led to the emergence of decentralized personal data management systems. Data semantics play a key role in order to enable both decentralization and integration of personal health data, as they introduce the capability to represent knowledge and information using ontologies and semantic vocabularies. In this paper we describe the SemPryv system, which provides the means to manage personal health data streams enriched with semantic information. SemPryv is designed as a decentralized system, so that users have the possibility of hosting their personal data at different sites, while keeping control of access rights. The semantization of data in SemPryv is implemented through different strategies, ranging from rule-based annotation to machine learning-based suggestions, fed from third-party specialized healthcare metadata providers. The system has been made available as Open Source, and is integrated as part of the Pryv.io platform used and commercialized in the healthcare and personal data management industry