Would it be possible to store their contents in an encrypted manner so that only the recipient would be able to read them?
I suppose that's quite possible, but not in the current default version of this add-on.
I'll take your suggestion into attention when taking my hands on it, as I have a couple of requests on more features to add, and adding an option to disable the "Sent" folder at a server-side settings or per-user level is not that difficult to implement (and it could be useful in terms of keeping server resources).
But as about adding an encryption to such messages... I'm not sure what is needed for? Even if we encode and then decode the message (and it must be encoded so the recipient could read it), all mechanisms are stored on the same local server and are possible to break-out if the server files are stolen. It's not like about Telegram or Whatsapp messaging when there are different security keys for each client and which that's why are hard-to-break.