You don't have anything in your outbox because the option is disabled by default.
If you want to enable it, go in settings, General settings and check the option:
Sent Private Messages: Save a copy of sent messages in my Sent Items folder by default
As for nobody replying to you, they probably simply didn't reply. It does happen that a message doesn't reach it's target, but it's pretty rare in my experience.
I can guarantee private messaging works since I use it fairly often.