
Personally I don't really care too much. As long as the material is found on my site I'm excited about it. Yes it's fun to be the person revealing whatever that material is, but you're also the one prone to screwing up or damaging that material as well. I'm not saying that I don't want to be the individual finding things because I'm afraid of destroying the remains, I'm saying "hey thumbs up buddy". Archaeology is a destructive profession. You're destroying the record, and probably making mistakes you won't realize until later, or perhaps mistakes that others will realize lifetimes later. That's just the way it is.
Another reason there is competativeness in archaeological digs is because people want something to work on, like a goal. Sure, there is the fact that you are trying to answer questions about humanity, but when it comes to the labour, people want to be the guys who found the most artifacts, or found the best preserved building, or get to the bottom of their unit the quickest. Not to mention the people who want to impress their professors: "Dr. Jones look! look! I FOUND THIS!"
Again, I don't care. And guess what? When digging in excavation units you are designated to that particular area. No shit right? No amount of skill or luck is going to change what remains there are, if any, below you. You will find things eventually, and you are no better than the guys in the next unit over because you found more artifacts. I want to know what these people were doing here, how long ago they were doing it, why they were doing it, and what does it tell me about their culture. Keep your head down and power through.