Good subject. I normally leave the paper in the safe. I should - and would move my paper to the freezer if it is to be kept for an extended period.
What types/ brands of Paper Safes are we using? Mine, after a few others broke, is an EPOI (?? - I'll have to check on that) "drawer type", feeding one sheet at a time when each individual compartment is opened. I love the thing (actually two "double 11 x 14s" stacked vertically).
This uses plastic "safeties" that clip onto the drawer doors to help in preventing using the wrong paper in total darkness.
What else is out there?