I'll guess that the sheets are old and getting porous, or the seal around the fasteners is breaking down, usually a bitumen mastic between the fastener and the sheet. If thats your issue and the sheets are sound, then undo the screws a few turns, remove any loose material under the square washers, put a dab of new mastic around the screw shank then retighten until its just snug. Its common for the sheets to be slightly damp underneath after extended periods of rain, this shouldn't be a problem. If the sheets are really porous, to the point that water droplets form on the underside, you are looking at a new roof, cleaning down and sealing old corrugated asbestos roofs often costs almost as much as replacement, and maybe much more if unscrupulous contractors pressure wash the sheets contaminating surrounding areas with asbestos fibre.
P.S. Before you walk on the roof, research safety on Fragile Roofs; there have been many serious accidents from people falling through these roofs