I did not think to take "during" photos, so you'll have to just trust me on this one. In order to improve sound control on this wall, we installed sound control insulation between the existing studs. I then drywalled, taped and mudded the wall. Finally, I framed a new 2x4 wall on our side of this drywall.
When we're ready to finish the room, we will install more sound control insulation between these studs, and then install drywall. Who knows how much quieter this room will be, but at a cost of 4" of space, $50 of materials, and 2 hours of time, it was worth a try.