Sure there is, most messages are probably too short but in general yes. There is no difference to an online article.

I write a book that gets published. I still hold copyright over it even if it is in someone else’s bookshelf. What rights the copyright holder and the person has is regulated by law. For example a physical book can be resold or lent to someone else, but it is not allowed to copy it and sell the copies.

I can cite text from the boom, that falls under fair use but I cannot use whole chapters in a derived work.

I still hold copyright over my messages online, even when it is public or published, that is basic copyright law in most relevant legislations. If the training of an LLM and later selling access to the LLM with copyright infringed data is fair use is yet to be determined.

You missed the point. It is not about if it is private or not, it is how they use it. You are allowed (on some pages) to read news article. Are you allowed to copy and publish them on your own site? No. You have a Copyright on your posts same as a author has on his books.

If it is legal or not is still to be discussed.

Similar to how data was mined (or even still is) about users without consent. Now there is for example the GDPR.