While main hand is right hand, bow can't be used in off hand (left hand).
MC-84419
Can't Use Bow In Off-Hand
Resolved
Duplicate
0
David Kordik
(Unassigned)
Unconfirmed
Minecraft 15w31c
Retrieved
[Mojang] Dinnerbone (Nathan Adams) resolved MC-82923 - Bow in Offhand does not work properly as 'Works As Intended'
You can't shoot from the offhand. The animation is wonky for using items unsuccessfully but that's its own bug.