Essentially, rather than having comments at the bottom, Medium’s “notes” are annotations inserted in the margins, and they’re private by default — authors must explicitly make each one public.
It’s a great idea. It still suffers from some of the problems of comments — for instance, cultivating the expectation that anyone’s thoughts deserve to appear to your audience and on your article — but by making them private by default, it removes a lot of negative incentives.
I’m still not sure I’d want them on my site. But I also wouldn’t write on Medium, so I’m probably not the target audience.