At one point it didn't (when Mail was using a local folder for sent mail), but currently it does. I had to use Settings to tell Mail which folder on the server "Sent" was the sent mail folder. I can't believe I had to actually do that, but at least the option is there.

To confim it wasn't related to moving messages around from Mac OS, I also sent some new test mail, and it also only shows my name in the list.

The other big thing missing from iPhone's mail client is any type of sorting. It's like the model T of mail clients. You can have everything exactly how you want it, if you want it exactly how it is. "Any color as long as it's black"