I've always drilled after etching because of two things.
1. The general opinion was to do it that way....and
2. After etching, there is already a guide 'dimple' for the drill bit to aid in centering on the pad.
I haven't done any double-sided boards yet, but I would envisage that in order to prevent damaging pads/vias on the bottom side of a d/s PCB, it is probably essential to use a brand new sacrificial backing board. One which is already full of holes can likely result in the drill bit being 'punched' through the PCB bottom side and creating pad tear-out/lifting, rather than cutting through cleanly.
EDIT: If anyone is in doubt of the merits of sacrificial backing boards, take a drill and make a hole in a piece of wood simply held in the hand. Next, place the same piece of wood atop another piece of wood that you don't mind destroying and make another hole. Compare the finish on the bottom side where the drill-bit exited the workpiece on both examples. The one with a backing board will always have a cleaner hole.