How outgoing do you have to be to be a pilot?

Having good one-on-one communication skills is very important in aviation. You don't need audience speaking skills. As a flight instructor, you'll be teaching and conveying ideas to your students. Once you get to a multi-crew cockpit, being able to get along with different people one-on-one is essential to making the flight successful. It also helps to make the flight go faster. Not talking on 4+ hr flights at all, except for required communication, will make it a long flight.

Strong communication skills are a great asset to what is referred to as Crew Resource Management (CRM) in the aviation world. You do not need to be a very outgoing or social person to have a succesful aviation career. However, being able to clearly communicate and resolve conflict is key.