Depends on which system you use.
A lot of CMSes have their own commenting system, so no need for an extra database because they’re all saved in the main database.
But sometimes you don’t want that, and you can use an external service (such as Disqus), which gives you more options, but the comments are saved on the Disqus server. Same with embedded Facebook comments (see this page for example) — in that case, even though it’s not a Facebook page and the comments are only visible on that specific page, all comments are saved on the Facebook servers.
