When I wrote Weblate I expected that major use cases will be free software projects, which will use it for public translating service. At least this was my intention for it.
But I seem to be wrong, at least for now. Most of people who contact me regarding Weblate are doing private installations and I already know few places where it was massively used. Unfortunately most of them rejected to be publicly mentioned, so I can't tell you the names. But they do exist :-).
Anyway I hope for some free software projects to adopt Weblate as well, otherwise I should have written it as commercial software :-)