Everyone else put their comments about general PvP. Get to know how to use the off-hand -- it's essentially an extra item slot and can save you a lot of time.
Generally, your best food should go in the off-hand (gapples -> steak/cooked pork -> other cooked meats -> other food). Your off-hand is versatile that it can be used for anything you need the most (e.g. need to block arrows => use a shield). I don't recommend a bow on the off-hand unless you're 200% sure you can land the shot or it's not FFA/duels (arrows disappear if you shoot them at the ground).
In short:
The advantages of the off-hand:
+You can use your items a little bit faster than switching main-hand slots and back.
+Acts as a 10th "hotbar" slot.
+Versatile.
Disadvantages:
-Cumbersome. You can accidentally use your off-hand when you don't want to.
-Unloading a loaded bow is harder. You can either shoot the arrow at the ground if you can recover arrows or you have to put it back in your main hotbar and load the bow back in.
-Enemies will know what you're holding in the off-hand. Sometimes it's better to hide certain items in your hotbar instead.