As the wiki of 1.9/spoilers of 1.9 update says, you could use a bow in left/right hand (depending on main hand), no matter what is the item in the main hand (unless sword). However, all it does while in the "not-main" hand when you right click is just "shake".

How to reproduce:
1. Put an item (that isn't a sword) on the main hand;
2. Put a bow on the other hand;
3. Right click, It won't shoot, even with arrows in the inventory.