EDIT: Check the suggestion I have after this one, as it is easier. This one could still work too, but requires a higher license level.
If you have an Advanced license (to check in ArcMap, Help, About), one workflow that could work would be this:
Buffer the line version of your polygon, being sure to set the "Line Side" parameter as Left or Right (may have to experiment to see which one works). Also, "Line End Type" should probably be "Flat" just to be safe (but probably doesn't matter).
ArcGIS Help (10.2, 10.2.1, and 10.2.2)
Note - this assumes the polygon was drawn with same directionality - i.e. all the linework was drawn clockwise or counter-clockwise. If that is not the case, you will have to change the line directionality or redraw the polygon as a line all in the same direction.
ArcGIS Desktop
Chris Donohue, GISP