Are e-mail addresses/names/message used in an e-card processed and stored only through my server, or is a third-party involved in the e-card process? I'm just wondering in terms of privacy, if the sender, recipient, and my site admins are the only ones who ever see or have access to this data.
The ecards are not being processed by any other server than yours. If you are concerned for privacy, I can assure you that coppermine doesn't contain a hidden email harvester routine or something along those lines.
Of course it lies in the nature of an email (and that's what the ecard is) that it is being sent through various servers on the internet, so a malevolent man-in-the-middle attack is of course possible. This applies of course to any email, not only ecards sent by coppermine. Everyone who uses emails should understand that private and security-related data shouldn't be sent by email; an email is as secure as a postcard in real life: the mailman can read it, and the mailman can make a mistake when delivering it.
HTH