9/4/2013
I have the same problem now....just started 9/1/2013..... on my iPhone 5. I am looking at shutting down the Settings - Sounds - Tweet tone to see if there is bleed-over from the type of message the device thinks it is receiving. There may be some app update which has caused the system to think it is a Tweet and a Message being delivered.
When this all goes wrong, it does the Tweet tone first; then the message appears, then another Tweet tone.
Testing Complete ----
Changing the tone on your Message (Tweet tone) setting gets rid of the double Tweet sound but, try just changing your Tweet sound acknowledgment first. If you don't use Tweet as a service, why have it do anything....try this as well.
This takes the quirks out. No new device required.
9/4/2013