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For robots to seamlessly interact with humans, we first need to make sure that humans and robots understand one another. Diverse algorithms have been developed to enable robots to learn from humans (i.e., transferring information from humans to robots). In parallel, visual, haptic, and auditory communication interfaces have been designed to convey the robot's internal state to the human (i.e., transferring information from robots to humans). Prior research often separates these two directions of information transfer, and focuses primarily on either learning algorithms or communication interfaces. By contrast, in this review we take an interdisciplinary approach to identify common themes and emerging trends that close the loop between learning and communication. Specifically, we survey state-of-the-art methods and outcomes for communicating a robot's learning back to the human teacher during human-robot interaction. This discussion connects human-in-the-loop learning methods and explainable robot learning with multi-modal feedback systems and measures of human-robot interaction. We find that -- when learning and communication are developed together -- the resulting closed-loop system can lead to improved human teaching, increased human trust, and human-robot co-adaptation. The paper includes a perspective on several of the interdisciplinary research themes and open questions that could advance how future robots communicate their learning to everyday operators. Finally, we implement a selection of the reviewed methods in a case study where participants kinesthetically teach a robot arm. This case study documents and tests an integrated approach for learning in ways that can be communicated, conveying this learning across multi-modal interfaces, and measuring the resulting changes in human and robot behavior. See videos of our case study here: https://youtu.be/EXfQctqFzWs
