The companion ticket is a discount code. Once ticketed, you each have your own tickets on the same reservation, just like any other purchased (with cash or miles) ticket. You can check in together or separately. The companion is on a regular ticket in a regular booking class (with full frequent flyer elite privileges and earning on AS or partners based on the booking class, etc). The only difference between buying a reservation for two travelers with the companion fare and without the companion fare is the amount charged to the credit card.

The companion code cannot be returned, which (by my understanding) is the main mechanism for preventing you from misusing it by not traveling together. (If you want to cancel or change one ticket, fine; the companion fare is used and can't be used again, and you'll have to pay the fare difference up to the full going rate at the time of the change, plus change fee.) I have never been asked about my traveling companion when boarding or checking in, and we don't always board together.