Communication bandwidth is an important consideration in multi-robot exploration, where information exchange among robots is critical. While existing methods typically aim to reduce communication throughput, they either require significant computation or significantly compromise exploration efficiency. In this work, we propose a deep reinforcement learning framework based on communication and privileged reinforcement learning to achieve a significant reduction in bandwidth consumption, while minimally sacrificing exploration efficiency. Specifically, our approach allows robots to learn to embed the most salient information from their individual belief (partial map) over the environment into fixed-sized messages. Robots then reason about their own belief as well as received messages to distributedly explore the environment while avoiding redundant work. In doing so, we employ privileged learning and learned attention mechanisms to endow the critic (i.e., teacher) network with ground truth map knowledge to effectively guide the policy (i.e., student) network during training. Compared to relevant baselines, our model allows the team to reduce communication by up to two orders of magnitude, while only sacrificing a marginal 2.4\% in total travel distance, paving the way for efficient, distributed multi-robot exploration in bandwidth-limited scenarios.