The first warning that the shower drain upstairs is leaking may be water spots of the ceiling, and those spots can be the precursor of drywall and even framing damage if you don't fix the drain. It may seem like a daunting project, but likely isn't. Leaks are often the result of dried, cracked sealant letting water in at the joints between pipes and drains. You can take the drain apart yourself and save yourself the cost of a plumber.
Unscrew the strainer with a Phillips screwdriver and pry it off with a flat-head screwdriver. If you don't see screws, you should be able to just pry the strainer off.
Wedge the flat-head screwdriver against one of the crosspieces inside the drain assembly and tap the screwdriver with a hammer to force the drain to turn counterclockwise. Keep tapping until the drain is loose enough to unscrew and remove by hand or with the help of adjustable pliers.
Clear away all the dried plumbers putty from around the drain opening with the screwdriver. Reach inside the drain and pull out the two washers just under the shower floor. Squeeze them together to get them out of the drain.
Take the drain and gaskets to the hardware store to purchase replacements. It's a good idea to replace the drain because it may have a hairline crack that you can't see, and that may be responsible for the leak.
Place the black rubber washer on top of the white friction washer, squeeze them together and push them into the drain. Fit the washers back into place, just under the shower floor.
Spread a generous bead of silicone caulk around the drain opening and screw on the new drain. When you've hand-tightened it, wipe off the caulk that has oozed out around the edges with a paper towel. Tap the drain clockwise with the flat-head screwdriver and hammer to completely tighten it, then wipe off the caulk again.
Tap the strainer back onto the drain with a hammer. Replace the screws, if necessary.