That’s how it already works. if you try to link to a thread on a site set to private, it will direct you to the login. Here’s what it looks like on mine (we use SSO so it’s not Discourse default.)
https://forums.nanowrimo.org/t/got-buddies-now-what/314851
It won’t pull the OpenGraph stuff because it’s private and it can’t see it. You’ll have to hand-code that if you want it to appear.
There is no way I know of to flag individual discussions to ignore the site’s settings for privacy. Even with page publishing, it respects the privacy of the source.