Does this store the emails my site sends?

No, not the messages themselves — not by default.

What is kept is who a message went to, what it was called, which plugin asked for it, when, and whether the mail server took it. The body is only stored if you switch on Keep the message itself as well under Settings → What is kept.

There is one exception: a message that was refused and is waiting to be sent again has to be kept in order to be sent. That copy is deleted the moment it goes, or the moment it is given up on. The daily tidy-up gives up on a waiting message when trying again is switched off, when it is older than the number of days you keep the log for, and when its last attempt was cut off part way, once it is a day old. With Keep the message itself as well on, every copy stays for as long as its row, this one included.

When somebody asks for their data, Tools → Export Personal Data includes every message the log holds with their address on the To line or in a Cc, Bcc, Reply-To or From header, and Erase Personal Data deletes them.