The British Medical Journal has an article that makes the point, including quotes from Tal Zaks of Moderna, that the studies aren't designed to be statistically significant indicators of disrupting transmission, preventing hospitalisation, or even preventing death:

https://www.bmj.com/content/371/bmj.m4037

So, from what I can tell. It isn't proven to disrupt transmission or reduce hospitalisation/death. It would require much larger or longer studies, which they have opted not to do.

I think lesswrong is wrong here. I looked at the document that lesswrong links, and I couldn't find any statement about effectiveness of the vaccine in preventing infection after two doses.