Communication software such as Clubhouse and Zoom has evolved to be an integral part of many people's daily lives. However, due to network bandwidth constraints and concerns about privacy, cameras in video conferencing are often turned off by participants. This leads to a situation in which people can only see each others' profile images, which is essentially an audio-only experience. Even when switched on, video feeds do not provide accurate cues as to who is talking to whom. This paper introduces GazeChat, a remote communication system that visually represents users as gaze-aware 3D profile photos. This satisfies users' privacy needs while keeping online conversations engaging and efficient. GazeChat uses a single webcam to track whom any participant is looking at, then uses neural rendering to animate all participants' profile images so that participants appear to be looking at each other. We have conducted a remote user study (N=16) to evaluate GazeChat in three conditions: audio conferencing with profile photos, GazeChat, and video conferencing. Based on the results of our user study, we conclude that GazeChat maintains the feeling of presence while preserving more privacy and requiring lower bandwidth than video conferencing, provides a greater level of engagement than to audio conferencing, and helps people to better understand the structure of their conversation.