OK, thanks to diwic, there is a known workaround.
1. Install the "Pulseaudio Volume Control" application from Software Center
2. Open "Pulseaudio Volume Control" from the Dash
3. In the Input Devices tab, select "Analogue Input" in "Port:"
4. Deselect the padlock so that the channels are not linked together
5. Drag the Front Right channel all the way to the left: 0% (-∞dB)
6. Set Front Left to whatever mic volume you choose (e.g., 100%)
Skype and Mumble should now work fine. The microphone volume control which appears in the Sound Menu will also now work to control your mic volume in Skype/Mumble (it controls only the Front Left volume when Front Right is 0%).
This doesn't fix the bug, per se, but it does let affected technical people fix it for themselves.