Typically the process is as follows:
1. A site announces it is shutting down.
2. Archive Team discovers the site is closing and adds it to the Deathwatch.
3. Members of Archive Team collaborate to write scripts to extract user data from the website, using an existing archiving framework [3] developed for / by Archive Team.
4. Members of the public are invited to run instances of the archiving software [4], resulting in a distributed archiving network. This is often quite powerful, and has to be managed carefully to avoid overloading the site being archived.
5. Once completed, the data is (in many cases) uploaded to the Internet Archive to be stored permanently. Note that Archive Team is not affiliated with the Internet Archive, although we greatly appreciate their willingness to host this sort of data.
I'm not particularly actively involved with Archive Team at the moment (too busy with real life!) but I was heavily involved with a few projects, the most significant being the project to archive Yahoo Groups in 2019, which was quite complex due to the site structure (and Yahoo being less than helpful [5]). We got a lot (much more than I ever expected to get) but the majority of all that data was deleted.
[1] http://wiki.archiveteam.org/
[2] http://wiki.archiveteam.org/index.php/Deathwatch
[3] http://wiki.archiveteam.org/index.php/Dev/Seesaw
[4] http://wiki.archiveteam.org/index.php/ArchiveTeam_Warrior
[5] https://arstechnica.com/tech-policy/2019/12/verizon-reported...