If a message doesn’t show at inbox it is nearly unusable. There is one reason why I (or someone else) am using that: drafts. Everytime when some is asking if there any chance to save more than one draft the answer is send a PM to yourself.
UX-wise sended PM must show up on (at? I really dislike prepositions 
) inbox, even when sender is same as recipient. And when a user uses a lot PM finding relevant message is much harder when it is burried in sendbox.
Sure, this is not high priority question. But it should be fixed at some point and then there is just two options:
- recipient can’t be same as sender (and that is not an option, if I’m right, because it will break CCs of group messages too?)
- sended message goes to inbox as expected
There is door #3 too: allowing more than one draft 