Thats how the software is designed, it will only save to the sent items folder on the device if the email has been composed and sent with the device. You cant send something in outlook and expect it to be on your device sent items folder.
However, if you are sending with the device and its still not showing up, there might be a problem. Honestly I would try re-registering the account before anything. But if its still not working, let me know what device you have and maybe we can troubleshoot it.