I found a way to address this, although it's not a 'fix.'
While the sound is playing, you're able to press the volume buttons and it will change the volume of the skype notifications. This is no good for when the 'message sent' sound is playing, because it's too short to work with, but the 'sign-in' sound is nice and long.
So the process is this: double-tap the home button on your iPad, which will bring up the running apps menu. Press and hold skype until it starts wiggling and there's a red minus on the upper-left, and hit that minus. That will close Skype. Press the home button two or three times slowly to get back to the home screen. Launch Skype. While it signs you in, you'll hear the sign-in sound, which is nice and long, and during which time you can press the volume down button to lower the volume. Repeat as necessary. I had to do this three times to get the volume down as low as I wanted, but it does work.
I haven't received any messages since, but my experience was that receiving Skype messages while the phone was in standby was at a normal, low volume but then I'd send a reply and that notification was way too loud, so I have to assume that this will affect it.
It would be nice to simply turn off those sounds altogether though, since I do have no need to be notified that I just sent a message...