Clickbait / misleading title - these aren’t private chats:
> says an Anthropic spokesperson, “These shareable links are not guessable or discoverable unless people choose to share them themselves. When someone shares a conversation, they are making that content publicly accessible, and like other public web content, it may be archived by third-party services.”
These are publicly shared convos, turning up in search engine results - just like every other publicly available content the search engine had crawled/indexed.
This is the system working as intended / expected
That might be a case where what you expect is private is leaked by the browser