They're running locally. Stallman has a pretty good essay[0] on this, where he calls out SaaSS (Service As A Software Substitute) because the majority of SaaSes are perfectly capable of being self-hosted, operated peer-to-peer, or locally stored. Obviously that doesn't make money though, so people draft up a price and offer convenient liability to business users (like "Free Lifetime Support for just $999 more a month!"), which leads to the present shitshow that we're living in.
[0] https://www.gnu.org/philosophy/who-does-that-server-really-s...