Kinda puzzled by this. Who's out there posting to the open Fediverse with no expectation that their posts are going to be seen, indexed, and possibly archived? It's like publishing a web page and expecting that search engines won't index it.
Just like with other services, you should expect that anything you post to the Fediverse will be there forever, tucked away in databases, indexed by search engines, and outlive you. Even more so, since editing or deleting Fediverse posts is more like a polite request than an actual command; some servers ignore such requests even if your home server complied.
That people are getting upset at this idea gives further credence to the idea that these newcomers really have no idea what they're joining up to and are just hoping that "Mastodon" matches the ideal Twitter replacement they have in their heads with little to no understanding of the reality of its workings.