My experience of two captains flying together comes from two different sectors of the industry, both with hazards. First, in a non-public transport operation, flight crews flew as Capt / Co-Capt with command rotating on each leg. Due to the nature of the operation, only a few captains were qualified to command particularly demanding flights. With a very small group the procedures were standard (if not always written down) and personal interaction was both strong and balanced. However many slips and mistakes were made, most were quickly caught and corrected, but complacency and failure to behave as a NFP led to a few interesting events.
My second experience is from investigating incidents and accidents. I saw evidence that flights with two Captains significantly increases the risk of an event. Even worst three captains on the flight deck, whether or not the jump seat pilot was in official or unofficial capacity (a caution here for check flights and LOSA, but of course everyone is flying ‘by the book’ on these sectors).
The evidence indicated that a captain NFP is not a good monitor, this also applied to a lesser extent to training captains during check flights. A monitoring captain intervened much later than would have occurred with a recently qualified first officer. Some pilots did not intervene at all. The reason for this appeared to be the assumption by the NFP that the handing pilot knew what he was doing, or at least as long as the monitoring pilot knew what was happening he assumed that he did not have to alert the Commander. This may be similar to an Instructor letting the student go too far with an incorrect course of action.
With two captains, their joint ability to detect threats and hazards appeared lower than with a balanced authority gradient; thus during an event with two captains they were at higher risk before recovery action was commenced. I wonder if this is similar to group risk theory; where a decision taken by a group of people tends to be more risky than when taken by an individual. Although CRM encourages group participation, the responsibility for any decision rests with the commander. With a level authority gradient, two captains may operate more as a group, whereas with a balance authority gradient this maintains the respect for the commander. These aspects are seen in other industries, particularly at board level.
Another aspect of incidents and accidents was that a monitoring captain was not familiar with the duties of that role, particularly standard calls and physical actions from the right seat. The lack of intervention by a captain was also apparent when there was a first officer handing; errors by the captain in these circumstances were more biased to the lack of awareness and complacency; CRM aspects.