How can I organize my classroom so that students can work in teams, but still have their space?

This approach works if you have individual desks. Organize desks in paired rows that face the front of the room. During blackboard or overhead projector activities student can face forward and won't be distracted by teammates. When it's time for teamwork, have students group into fours by simply adjusting their chairs. See the diagrams below.

Activities Focused Around Blackboard and/or Overhead Projector Information

Activities Centered Around Teamwork