If you're referring to the swoosh sound when sending a mail, you can select 'None' in 'Sent Mail' under the sounds settings. However, if you're talking about the messages app then no, you can't turn it off. If I remember correctly though the messages app doesn't have the swoosh sound.
Um, yes it does. You only hear it if you have the phone not on silent though. I found that if you go to settings notification center messages alert sound and set to none the sound in the app will be gone, but you also won't get a sound outside of the app.