Data sharing is the fuel of the galloping artificial intelligence economy, providing diverse datasets for training robust models. Trust between data providers and data consumers is widely considered one of the most important factors for enabling data sharing initiatives. Concerns about data sensitivity, privacy breaches, and misuse contribute to reluctance in sharing data across various domains. In recent years, there has been a rise in technological and algorithmic solutions to measure, capture and manage trust, trustworthiness, and reputation in what we collectively refer to as Trust and Reputation Management Systems (TRMSs). Such approaches have been developed and applied to different domains of computer science, such as autonomous vehicles, or IoT networks, but there have not been dedicated approaches to data sharing and its unique characteristics. In this survey, we examine TRMSs from a data-sharing perspective, analyzing how they assess the trustworthiness of both data and entities across different environments. We develop novel taxonomies for system designs, trust evaluation framework, and evaluation metrics for both data and entity, and we systematically analyze the applicability of existing TRMSs in data sharing. Finally, we identify open challenges and propose future research directions to enhance the explainability, comprehensiveness, and accuracy of TRMSs in large-scale data-sharing ecosystems.