This is relatively new, but sites on Discourse’s Basic, Pro, and Business plans can all be hosted on a subdomain of discourse.group. Custom domains can be configured for sites on the Pro and Business plans. Sites on the Basic plan have to stick with a subdomain of discourse.group: Discourse pricing | Discourse - Civilized Discussion.
Discourse is open source software. It can either be hosted by a hosting company (for example, Discourse’s hosting business) or self hosted. In either case, you are still using the open source version of Discourse. I’m a little biased, but I’d choose to have Discourse host my sites for me whenever possible.