No, your analogy is not accurate. If Facebook and Twitter were part of the Fediverse, you might be able to post to one from the other, or you might not, depending on whether one had defederated from another other or not.
To extend the poor email analogy, it would be as if you had a Gmail account and tried to email a friend on Outlook, but you couldn't because Outlook refused to accept emails from any Gmail address, but you could get through to them if you sent it from a Yahoo address instead.
Aren't "private messages" not really private on lemmy anyway?