I almost always defer to others. That's what happens when you're the oldest of 4 siblings that all hunt. With my buddies there is almost always 3 of us, so I get to go my own way and the other 2 stick together. They generally use option 2.
As others have said, it is all about what everybody has agreed to. As long as there is a clear plan in place, then nobody gets their feelings hurt. I love helping others get their animal, as long as they are capable shooters. This year I helped 3 people fill their tags while I had a tag in my pocket as well. Two of them got their first elk and the other got his third mule deer. It is all about everybody's expectations.
Who ever spots it gets first shot or choice of deferring to another shooter. In some circumstances there will be a designated shooter regardless of who spots it. But you better make the first shot count because after that it's fair game if someone misses unless again that person is the designated shooter.
If it's around home it's whoever shoots first. Usually we are calling coyotes or pushing timber for deer at the end of season if we are not hunting solo. Not conditions that work well for taking turns. If we are taking a kid or someone after their first animal we adjust.
For out of state hunts I always go with the same guy. We flip a coin before the hunt starts and the winner of the coin toss can take that day or defer to the next day. Once the winner picks we alternate days. Whoever's day it is has first right of refusal on game that is stationary and looks like it will be for a while that day. If it's a matter of shoot or let it get away, rules are off and whoever can get it shot shoots. Once an animal (elk) is hit if it doesn't go right down both of us will try to get another bullet in it.
We have hunted together since we were kids. We tried everything from trying to shoot as fast as possible to beat the other guy to counting down and shooting at the same time. Nothing worked that great. This system has made us more successful because we take unhurried shots and take time to get a steady rest if at all possible. The guy not shooting helps with ranging and checking wind if necessary, plus trying to spot the result of the shot. If only one of us is successful on a shared hunt we split the meat.