There is no obligation to apply through an agency for European countries, except for the Netherlands and Switzerland (this only applies to au pairs who need a visa). Citizens of the European Union benefit indeed from the right to free movement. You are, however, obliged to apply through an agency if the government of a country stipulates that au pairs and host families need to organise the au pair stay through a designated agency. Such an obligation only exists for the USA, Switzerland and the Netherlands.
In the USA, the government has stipulated that au pairs and host families must turn to one of 16 designated agencies in order to organise their legal au pair stay. The agency helps with the application of the required J1 visa. Most of the agencies have associate agencies in the corresponding host countries. Au pairs and host families need to pay a fee for the services offered by the agency.
Why do I still find American host families and au pairs who wish to go to the USA on AuPairWorld? Au pairs and host families search and find each other on their own on AuPairWorld. This principle also applies to the USA as a host country: host families and au pairs can make their choice according to their own criteria (prematching) as a first step. In the second step, they must agree on one of the designated agencies. You can find more information in our section Info host countries and the interview with Ayusa-Intrax: "How to become an au pair in the USA".
In Switzerland, au pairs from non-EU countries need to pass through an agency designated by the State Secretariat for Economic Affairs (SECO).
In 2010, the Modern Migration Policy bill was approved in the Netherlands. The Act, which is due to come into effect on 1 June 2013, will introduce major changes regarding the au pair stay. Dutch host families and their future au pairs will have to turn to a recognised au pair agency in the Netherlands, a so-called "sponsor", so as to submit the au pair application and handle the necessary administrative paperwork.