Co-Drivers come in many different categories and I suspect each have different sets of rules by which they either choose to abide or are required to abide depending on the terms of their work agreement.
You have an owner with a co-driver, a driver with a co-driver, or co-owners that are co-drivers. Co-drivers start out as either strangers, aquaintences, friends, spouse and spouse, or other family members. Whatever the case, a good understanding of business or personal decision making is essential or the driver / co-driver relationship will sour until the point they become adversary / co-adversary.
Since you want a first person account of a co-driver relationship, I'll play the role of co-driver as I usually do with my wife Rene'. I agree that, when time requires, the person alone in the truck, or alone on the phone, makes the decision to accept or reject a load without input from the absent partner. I agree that when my partner makes a bad business decision, while I'm absent, I'll abide by that decision. I agree that any heated discussions of personal matters, at the moment a run offer is tendered, will immediately be suspended and I will adopt a business demeanor in discussing a run offer.
I further agree that my partner must satisfy her chromosomal urges to go shopping while on layover and I won't attempt to control those urges until such time as we have no room in the van for freight.