In an age of mass communication, citizens need to learn how to detect and transmit reliable scientific information. This need is exacerbated by the transmission of news through social media, where any individual has the potential to reach thousands of other users. In this article, we argue that fighting the uncontrolled transmission of unreliable information requires improved training in broad epistemic integrity. This subcategory of research integrity is relevant to students in all disciplines, and is often overlooked in integrity courses, in contrast to topics such as fraud, plagiarism, collaboration and respect for study subjects. Teaching epistemic integrity involves training epistemic skills (such as metacognitive competences, capacity to use helpful heuristics, basic statistical and methodological principles) and values (such as love of truth, intellectual humility, epistemic responsibility). We argue that this topic should be addressed in secondary school, and later constitute a fundamental component of any university curriculum.