If you want nvidia video card and amd/ati video card in the same computer, you need to install both drivers.
Both video cards will work happily in the same system. Each card will render only to their own respective attached monitor(s) though. You could run 2 monitors with 1 connected to each video card and play 1 3d game on each monitor and both gpu's would be working on their own respective monitor. No idea what would happen if you dragged a windows 3d game between 2 monitors though.
Of course if your interest is gaming on 570 and mining on 5970, the same will apply. Install both drivers.