The Swedish Migration Agency writes about employing au pairs:

An au pair is a person who lives with another family and is paid for child minding and light housework . The purpose of their stay in Sweden must be to obtain international experience and the opportunity to learn Swedish language and culture.

And even though there is no official au pair program in Sweden (Sweden did not sign the "European Agreement on Au Pair Placement") there are special rules for certain occupations and citizens of certain countries.

'The work of an au pair is such an occupation where special rules apply.

Work permits for citizens of Australia, Canada, New Zealand and South Korea allows young citizens of these countries to 

• travel and work in Sweden
• take up incidental employment
• progress in their chosen careers through employment in Sweden that is related to their degree and / or professional experience
• improve their knowledge of the Swedish language
• get to know the Swedish culture and people 

More information can be found in section "Work permits for citizens of Australia, Canada, New Zealand and South Korea" on the website of the Swedish Migration Agency 

Note:

Sweden did not sign the "European Agreement on Au Pair Placement".

An au pair being an EU/EEA-citizen has the right to work, study or live in Sweden without a residence permit. This is the so-called "right of residence" as long as the au pair has sufficient means to support him - / herself.

Regarding the work as an au pair the Swedish Migration Agency points out:

An au pair is a person who lives with another family and is paid for child minding and light housework. The purpose of their stay in Sweden must be to obtain international experience and the opportunity to learn Swedish language and culture.

Note:

Sweden did not sign the "European Agreement on Au Pair Placement".

More detailled information about the "Free Movement" of EU Nationals  and what it all includes can be found on the website of European Commission in section "Free Movement - EU nationals".