I only hunt with one other person... Too many people, IMHO, lends itself to too many chances for someone to "oops" and scare off game. My hunting partner has been at it way longer than me, so he always insists I get/take first shot, so I can have the same thrills he has had. His mindset is, new hunters or folks that haven't had much success go before him.... funny thing is we always get in back-and-forth discussions who is the "shooter'. I should also add that this only occurs when we are traveling to or from a hunting spot, as when we are sitting in blinds or stands, we are far enough apart that only one of us will have a shot. Truth be told, he is usually the "shooter" when traveling from hunting spot to camp.... I hate the way he drives, and since we generally take my truck, I don't want him driving my truck!
Lastly, when we are hunting, it's all about a "we" experience. It really doesn't matter who harvested game that day because we both worked for it, so we both harvested. I am always beyond excited when he takes game, as he is for me. Not to mention, success isn't 100%, so we make it all about the time in the field with each other and comparing notes on what we observed/learned. The only time we will go with others is when we guide for "Hunts for Heroes". Then we make it a little competitive and try to get our hunter to succeed before the other.... yet still work as a team to locate and find game and celebrate the lucky hunter to harvest.