As pointed out by Chance Miller, Apple has published a new white paper about the privacy it has built into the Vision Pro. Here is a bit from its introduction:
We built visionOS from the ground up to protect your privacy and the privacy of people around you. Apple Vision Pro is the first Apple product that uses advanced always-on camera streams of your eyes and the world around you to enable spatial experiences. And where you look can reveal what you are thinking, such as links you almost clicked or apps you thought about downloading. To keep your thought process private, where you look before you interact with content is not shared with Apple or the apps you are using, and does not leave your device. Similarly, because Apple Vision Pro blends digital content with your physical space on-device, the apps you use cannot access information about your surroundings by default.