It is not a pleasant situation for both of them - the passenger or the large passenger. It is very uncomfortable for the passenger and totally embarrassing for the large guy. In addition to that, it is uncomfortable and embarrassing for the staff.
Unfortunately, there is no one single rule that applies here; each airline has its own policy regarding this.
What you should do before the flight takes off
If the large person's body is invading your space and you have noticed this during boarding, make sure to speak to one of the cabin crew members immediately. Make it their problem. You are paying for a seat and it is your right to get a full seat, not a part of one.
Remember, the only time you can report this case is on the ground. It is the perfect time for crew or ground staff to find another seat either for you or for the large person, or to find another solution in the case that there are no spare seats. Once the doors are closed and the plane is moving without you talking about it, you have given complicit agreement to this situation and no one can do much about it in most cases.
What the staff will do
This depends on the airline. Some airlines make large people pay; some others upgrade large people to business or first class for free as a part of their services (the airline I work for does this if no two seats are available in economy class). You could get lucky and get moved to a higher class and leave your seat for the other passenger. Either way, your problem should be solved before the plane takes off.
Things to keep in mind
This is an odd and embarrassing situation. Make sure to talk in private with the cabin crew to avoid embarrassing the sized passenger publicly. Depending on the airline, the sized passenger can be deplaned from the airplane or asked to pay for an extra seat. You may also be asked if you would like to wait for another flight! Anything is possible if no extra seats are available.