here are shots of the bows i'm using for this test -
bows both the 'regular' and shadow bows are comparable, though the shadow is slightly better. the bone is a lesser quality, but used to demonstrate.
i took the bows, saved, and (re)ran the same monster being attacked with each.
there are 3 pics of (reloaded monster) regular bow (just piercing damage)listed as piercing 1,2 and 3, three using the shadow bow (piercing AND shadow damage) listed as shadow 1,2 and 3, and 1 with the bone bow (piercing AND bone) listed as bone1.
zregbow and zshadowbow1 & 2 are against an EE.
bow damage note that all contain some fire damage (from something else) and that the piercing damage of the regular bow is significantly more than from the shadow or bone bows.
the regular bow shows damage from 55 - 73 points piercing.
the shadow bow shows damage from 31+46 to 31+58 (piercing + shadow).
the bone shows 22+39, but remember, it's of inferior quality.
all in all it appears that there is indeed
some extra amount of damage utilizing a shadow bow over a traditional bow. this is probably more pronounced dependant upon the resists of the monster.