Hi there. I have a 2 yr old and a 6 yr old as well. My 6 yr old is very kind and empathetic. My 2 yr old was born running, jumping, hitting, and ready for life.
My point - it may just be some kids personalities to be ready to get the most out of life and don't want to be slowed down by anyone. I don't think it's a matter of being mean, I think this is just how they get results.
What do we do about it and what can I recommend? Maybe not a whole lot. We try a lot of positive reinforcement of good behavior and time-outs for bad behavior. I don't want to start spanking her especially since hitting her as punishment for hitting seems counter-intuitive. We try modeling the behavior we want and enjoying the many bright and wonderful sides to her. My next step is books for spirited children.
The greatest thing that I have found with my niece, my friend's son, and my own (all kids with similar temperments) is that the joy and laughter they bring is amazing. My niece is 8 now and her behavior has greatly improved. She still has talking and moving around issues in class. I don't think that will ever go away, nor should it. It will probably be energy that moves her ahead in life. We just need to learn how to teach our kids that its ok to be who you but you have to treat people right and follow the rules in classrooms, at home, on the road, and eventually at work, in society, etc.
I hope this helps, even if it's not a whole lot of advice. Our bottom line is a lot of love and acceptance matched with a lot of reinforcement of our rules for acceptable treatment of others.