It seems that on OS X there is a problem with menu items not getting enabled/disabled appropriately.
At least for your use-case, you can just copy/paste your tiles by using right-click-drag to capture the tiles and then left-click to paint with them elsewhere. You rarely need to actually use the copy/paste feature, unless you want to copy a piece of the map to another map.
That certainly seems like the issue I’m seeing and am using version 0.11.0 but I’m running OS X 10.10.3, if that makes a difference.
Unfortunately, when I right-click on a tile and drag nothing happens (no selection made or cursor change). Selecting first and then right-click dragging doesn’t produce any results either.
I downloaded 0.10.2 and copy/paste works, but I performing right-click drag to capture and paint tiles produces no visible action. I’m new to the Tiled application though so maybe I’m missing a step.