Despite the continuing efforts to improve the engagingness and consistency of chit-chat dialogue systems, the majority of current work simply focus on mimicking human-like responses, leaving understudied the aspects of modeling understanding between interlocutors. The research in cognitive science, instead, suggests that understanding is an essential signal for a high-quality chit-chat conversation. Motivated by this, we propose Pˆ2 Bot, a transmitter-receiver based framework with the aim of explicitly modeling understanding. Specifically, Pˆ2 Bot incorporates mutual persona perception to enhance the quality of personalized dialogue generation. Experiments on a large public dataset, Persona-Chat, demonstrate the effectiveness of our approach, with a considerable boost over the state-of-the-art baselines across both automatic metrics and human evaluations.