Many many years ago, when I went to school in England, the headmaster was known as a beak. They used to call judges beaks.
I don't know how it got started on the rivers, but from my experience, it's all about territory and intimidation. If you show up alone and occupy space where a group of good buds have been since first light, you could be branded a beak.
Add a flyrod and all the roe guys now know for sure you are a beak. If you hook something and they are all skunked,you may want to move on, so great will be your beakdom.
In all the areas on the river that can be fished from both sides, the guys on the other side (whichever side it is) after a few crossed lines are beaks. It's quite easy to figure out. If you have a good spot to yourself and someone comes along, he's a beak. Go on - admit it, were all the same