I hope someone responds to you because I am having the same experience. I had an iPhone 3 and liked the sound when I received a new text. I only get the puzzle piece, no sound, with the Captivate. As a new (very challenged) user, I am hoping this site will shed some light on the functions I have not mastered yet.
OK, here's what I found out. After removing the battery, finding a reset to factory defaults, I got the little blip default notification ringtone back. It would not take my custom ringtone or any preset ringtones. In other words the default ringtone is all the sound it would make. AT&T store couldn't figure out what was wrong. Sent me to get a new phone under warranty. At the warranty store they spent about an hour and finally located another place to set the ringtone sound. Normally and originally I set the ringtone I wanted under Settings, Sound and display, then set the tone under Notification ringtone. That is the way everyone was trying to get it to work.
The other setting that they found was under Messaging, Options, Settings, scroll down to Notifications, make sure it is checked and then Select Ringtone (which is right below Notifications in the Messaging options. Now the ringtone is set to the one I want in BOTH places (Messaging options and in Settings). It works as it should now.
Let me know if that doesn't make sense and I'll try to explain differently.
Thanks for reading.