As far as I know, if the 'unwrap uvm' modifier is on top of the symmetry modifier in the stack, you can't apply different maps, cause one side is just a plain 'copy' of the other side. So, UV mapping this way causes exact the same UV-coordinates for both sides.
You can solve this by UV-mapping on top of the symmetry modifier (or symmetry beneath 'unwrap uvm', as in your screendump). Then collapse the stack and add another 'unwrap uvm' , select one of the two sides in the 'unwrap uvm' modifier and relocate it on the 0-1 uv grid. This way you can apply a texture map that has a part for the left side and another part for the right.
You can also detach one side after you uv mapped it onto the symmetry modifier. This way you can work with 2 different texture maps instead of one texture map for both sides ;-)