They finish unmasked so we can see their facial expressions. It wouldn't be especially emotional or inolving if we couldn't.
The Shadow is a wonderful movie. Although it originated as a radio show, it remains one of the best comicbook/superhero movies. It recaptures the sense of 30's and 40's pulp mystery so perfectly that it leaves stuff like The Phantom waaaaaaaaaaay behind. Alec Baldwin and Penelope Ann Miller are bullseye perfect in their performances, Baldwin does Bruce Wayne perfectly (this is the true Wayne from the comics, not Bale or Keaton) and Miller does a spot-on sultry screwball vixen.
With The Shadow, you absolutely have to know about the tone of pulp and radio serials to understand it. That's the problem with it.