Does the mirror modifier have to be applied in order to a complete a UV map of the model? I want to know if it's possible to have the complete UV map of an object while still retaining its mirror modifier. I have unwrapped the model, but it can only do so for the edited side. 

I'm using Blender 2.75.
It depends.  The Mirror modifier extends UV's as well.  So if you have textures mapped to your model, it will map to both sides identically.  If you are fine with that, then no.

If you need to edit both sides independently, you will need to apply the modifier and further unwrap your model.