According to Wikipedia, it varies. Some organizations don't require the jumper to pack their own parachute. What does seem to be a common principle is that "All riggers are jumpers and can be asked at any time to jump with a parachute they have packed." or in the US military, that any rigger has packed, without knowing who.
http://en.wikipedia.org/wiki/Parachute_rigger
Yep. It's not really that you have to pack your own parachute, more that it's cost and convenience to do so, so it's pretty universal as far as I am aware. I'm not certified or anything but I've jumped a few times and that's what the 10k+ jumpers said at the time.
