It all depends on company policy. At my previous employer, we were both typed and we each flew from the left seat. The pilot flying (left seat) was the PIC for that flight- listed as such, and had final authority.
At my current company, we switch seats (everybody is captain qualified), but the senior member of the crew is the final authority, regardless of which seat they occupy.
The reality is that given most circumstances and issues that I've had to face as a member of a two captain crew, we each respect the other's abilities and when a major safety of flight consideration needs to be made, generally the most conservative course of action between the two crew members is chosen.