A ticket and a reservation are not the same thing. Provided you each have one ticket covering all 4 flights then through checkin will happen (unless something really weird happens). This means that in both directions you will get boarding cards for both of your flights when you check in and your bag will be checked to your final destination. When you get to Paris your job is simple - find the gate for the connecting flight. Unless you get lost you don't need to go through immigration or touch your checked bags.
Enjoy Chile.