Place one nightstand upright on a work surface. Lay the other nightstand on the work surface with the back resting flat so the bottom end is facing you.
Apply a bead of wood glue to the bottom edges of the nightstand with the back resting flat against the work surface.
Lift the nightstand with the glue on the bottom edges upright and stand it on top of the already upright nightstand. Align the outside edges of the nightstands as you position them.
Remove the bottom drawer of the top nightstand. Drive trim-head wood screws through the inside bottom of the top nightstand into the top of the bottom nightstand every 4 to 6 inches in rows, using the screw gun.
Wash off any glue oozing out around the joint between the two nightstands, using the damp cloth. Insert the bottom drawer of the nightstand back in place.