Can a British Citizen Work In Any Part Of The Commonwealth or European Union Without A Permit?

I was wondering if a British Citizen would have very much trouble competing for a job in another country in the Commonwealth or European Union? I understand in the United States an immigrant has to obtain a Green Card and has to prove there is are not enough qualified applicants to fill the kind of work he is doing. Would a British citizen have to go through the same in another part of the European Union or commonwealth?