Mike, what do you mean by "noisey"? Is the problem squeeks and creaks or is it clumping from heavy footsteps?
Squeeks and creaks is from wood rubbing against wood or against metal fastners. This is usually warped boards doing the moving and nails are the common source for the other part of the squeek. Using screws with, or in place of, nails is a good way to solve this problem. You can also cleat the treads to the stringers with screws to stop the squeeks.
Because of the brittleness of drywall screws and deck screws, I would not totally depend on them staying together. Nails are more forgiving so a combination of nails and screws left in your stairs after rebuilding is appropriate. This is particularly true where the stairs attach to the rest of the house and nails would be my choice there. Nails are softer and will take more bending before breaking than drywall screws.
If your treads are significantly bending and moving as you clump up and down them, you might want to look at installing riser boards between the treads. This will stop or decrease clumping noises.