How do I turn off the text tone swoosh when sending text messages?

I can turn off all phone sounds, but is there any way I can specifically just turn off the send message sound?

Although it's not exactly what you might be looking for, if under Settings / Sounds / Sounds you set Text Tone to None, no sound will me made, neither when receiving nor when sending a Message.

I turned the default sound to just vibration, which turns off the swoosh sound and put personalized text alert sounds for individual people so I would be able to hear their incoming messages.

You can't turn it off, but there's a workaround on MacRumors forum. It involves accessing the phone system file and deleting the sound file.