I had exactly the same problem, but it wasn't with the sounds it was the vibrate, installed handcent, didn't like it, uninstalled and vibrate for texts and emails had disappeared, only thing that worked was a factory reset, havent installed it it since. hope this helps.

 

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Try going into settings on the stock messaging program and enabling the notifications. You may have turned them off originally so you did not get two notifications ?

 

I had the same problem with ringtones...

I selected a new one and not a custom one from within the program and it now works fine. If you removed software which was using a custom tone, the tones won't work.

 

I don't think it's a problem with Handcent as such. Handcent allowed you to choose any sound which the Desire won't do by default. I installed Rings Extended to let me select any audio file for a notification.