For movies, etc., my understanding is that it's P2P a la popcorntime.
P2P can be done for live sports via sopcast/acestream/etc, and (as far as I can tell) this seems more popular than the method I described above (eg: ripping content from live stream websites) for live sports, due to the quality difference.
In addition, P2P addons for movies a la popcorntime do exist, but the biggest addons that I have heard of: Genesis and Exodus, just rip content from file host sites.
But yes, if they're doing P2P, they're definitely liable to get letters. I think people overestimate the frequency of people getting caught for p2p file sharing, especially for live things that appear and disappear quickly.
Basically, they reach out to about 80 sites and scrape for links, then resolve those. It's a pretty insane process, but they've got it working just well enough.
The cost to run one of those sites is being overestimated here. They don't actually serve any of the video streams, just embedding from the dozens and dozens of little-known free live streaming services.