I feel like the real problem here is that the camera attached to your screen always seems to be centered above (or below), so you get a skewed looking-down-upon

I feel like the real problem here is that the camera attached to your screen always seems to be centered above (or below), so you get a skewed looking-down-upon-you situation - either you're staring at a forehead, or the other person looks like they're looking downward. A solution here would be mounting a camera on either side of the screen for a (very) wide stereoscopic view, and then averaging it in software.