To the best of my understanding, command points were one per player, not side. Same as commanders, so the situations would be as follows:
Command Points:
Kill your enemies' two, keep your two: +4, 1 point for each
Lose your two, enemies keep their two: 0
Commanders:
Kill your enemies' two, keep your two: +4, 1 point for each
Lose your two, enemies keep their two: 0
So across both, you can earn 8 points.
I believe the rules state that you must designate a unit to carry your command point, if it doesn't enter the field, it is count as lost, same as commanders.
Also, they don't get "used" I believe, more that if the unit carrying it is in certain situations, you can complete side objectives.