Would you guys add A private but public chat bot option so that it doesn't being shown on the library but It's only shown whoever shared the link of the chat bot.

ya mean a public character that is not listed/visible in the public character list.

As a work around, you can share the settings with whomever you wish so they can generate their own that is identical.

E.g. system command prompts, bio, temp/p weights, instructions.