Quote: " how do you achieve hand independence?"
Repetition and
lots of it.. For the hands, you'll need to build motor skills to control the sticks initially (Ie Grip). There are different aspects of the stoke also, from forearm, wrist and finger control. So were fine tuning the large to small muscle groups. Initially though, it tends to be easier to work on the large muscle groups first then as your grip/ ginger control improve the motion become more refined.
One of the first patterns we look at simply playing even subdivisions. 1/4 note to 1/8 note to 16 note. With a variety of stickings. Ie (Singles RLRLRLRLR, the reverse LRLRLRLR etc, Doubles RRLLRRLLRRLL, then reverse LLRRLLRRLL etc) and single handed. Once you get a handle on it, try triplets also.
It's important the reserve the stickings to built strength and control of your unnatural hand. Otherwise it'll become entirely dependant upon the lead hand. The same applies to your feet. Generally we'd play (such as the stuff above) a 1/4 note pulse with the bass drum or high foot foot (or combination) to increase the challenge and help us remove the invisible stings (puppet theory) between your feet and your hands.
Beyond that, another approach is to layer patterns over each other. So the right hand might player 1/4's while the left moves through the 1/4, 1/8, 1/16 then swap over. Then try the exercise with the right playing 8th notes, if left playing 1/4, 1/8 to 1/16. reversed the stickings etc etc. Layered patterns like this are also very useful exercises for developing flams,
This is effectively a permutation study, so picking up a snare drumming book that gives a good selection of sticking permutations can be invaluable (Stick Control, Ted Reeds Progressive steps to syncopation). Boring as hell though