In particular narrower topic research-level communities, that have a limited targetted audience and longer turning times for posts getting created than Q&A sites about less serious topics, have a hard time to successfully make it through the Area51 to graduation at StackExchange. This kind of high-level academic communities naturally does not provide the high amount of activity and traffic the StackExchange company considers to be a prerequisit for being continuously hosted. As a consequence, in the past a number of research-level science SE sites got closed either in public beta or after only a few days in private beta. The only still alive research-level science communities in the StackExchange network are MathOverflow (which was born self-hosted) and Theoretical Computer Science. So as I see it, the most important advantage of self-hosting (in particular high-level science) Q&A sites is that in this case, the community does not need to fullfill any by a commercial company externally prescribed activity and mass visibility criteria (compare Area 51 statistics).
Some additional advantages of self-hosting are
- Choosing an appropriate software and hosting, a basic rudimentary version of a Q&A site can be started much faster than going through the Area51 process and reach private beta.
- The community is free to determine its purpose, goals, rules, and policies and define its culture on its own, there are no externally prescribed guidelines, policies, rules, etc that need to be adhered to.
- Complete control over the (settings of the) software and features that get implemented. The StackExchange software gets continuously modified without any of the sites being able to reject new features or changes they dont consider to be useful for their local community.
Of course, depending on the ambitions self-hosting can mean in particular initially (much) more work than starting on Area51, the software choosen needs to be customised and maybe extended, the whole site description needs to be drafted and written, an appropriate styling must get developped and implemented, etc.
And most important: a self-hosted Q&A site needs its own technical administrators and system development team in the long run.
As soon as the site is more or less developped to the satisfaction of the community living on it and runs in a technically stable state, the time and effort needed to maintain it is not that big anymore.