There's no such thing as an IMAP mailbox. Mail receiving is split down the middle. Your SMTP server stores the email in a folder and your IMAP server provides access to that folder.
For context, I'm the author of Haraka, a popular open source SMTP server.
SMTP servers don't necessarily store anything. They have two responsibilities: Receiving and delivering email. What they do with the emails they receive is up to the configuration.
