I'm new, so this may sound stupid.
When I make a loop cut, I can see where it is on the original side, but not the mirrored side. Is there some option or setting I need to change in order to fix that?
There are two options to display the result of the modifier in edit mode:
(form the Blender Manual)
Box
Displays the modified geometry in edit mode, as well as the original geometry which you can edit.
Triangle
When enabled, the final modified geometry will be shown in edit mode and can be edited directly.
Enabling the triangle will allow you to "adjust edit cage to modifier result" :
Lemon is right. That option does allow you to view a made loop cut. However, when you are placing it and the pink line appears, you still cannot see the loop cut because the part of the mesh generated by the Mirror modifier does not yet exist. To make it exist, click the Apply button. You will not be able to change the settings any more, however.
The apply button:
Applying the modifier and the changes:
Lemon's notes in my words:
The triangle button allows for the generated portion to be visible in Edit Mode as if the modifier was applied. However, just because it looks like the modifier was applied does not mean it acts like the modifier was applied. Vertices will still be mirrored. Loop cuts will still only be visible in pink on the non-generated portions of the mesh.
The triangle button. Note how the mesh is still mirrored even though you can interact with more vertices. Also note how the pink is still only visible on half the sphere: