Thread:Extension talk:VisualEditor/Your wiki requires a logged-in account to access the API. Parsoid will not work for this wiki!/reply (3)

I can only answer half your question, but Parsoid is the backend system that allows VisualEditor to transform HTML markup to wikitext and vice versa. It is required for VisualEditor to function.

My guess as to why is doesn't work on a private wiki? I believe that Parsoid does not have any authentication schema in place. It can't access the wiki because it can not authenticate as a user - something required in a private wiki.