Yes, mounting tires on rims (even re-mounting) always requires balancing. Since practically nobody can mount their own tires, balancing is one of those things that "just happens" behind the scenes.
Most people I know who have snow tires opt for a second set of (typically cheap) rims, and the wheel set comes mounted and balanced. In my case, I change them out when needed, otherwise if I'm scheduled for something like an oil change, I just toss them in the car and have the shop do it.
They are always marked with chalk for which one is which (i.e., right, left, front, back).
