Until recently with openAI, I would not have assumed that sharing a chat privately with my friend would make it 'public'.

> If you share a conversation and google is made aware of the link

The issue, as I see it, is that google is made aware of the link without the user's knowledge. Its trivial to prevent Google from indexing /chat/* (or whatever the actual path is), and pressing 'share' and texting a url to a buddy does not immediately present itself as an action that Google would be aware of.

Nope. Doesn't work like that. The owner of a domain has full control over what is indexed and available on Google and other public search engines. It is their responsibility of keeping these links off the indexes by setting no-index meta tag / robot.txt / hiding it from sitemaps / blacklisting or removing it in search console.

Just because a link is publicly accessible does not mean it is publicly indexed.