The servers only store data that is needed.
You don't have to wait for a toot though!
Log in through the website or a third party app like Tusky, go to the user's profile and click their profile picture. This opens their profile in a new tab.
You can interact here remotely, or copy/paste URLs of particular posts (or profiles!) into search box on your own server. This forces your server to notice that post, and it will remain visible on your server in future.