There is a free labour market in the EU/EEA/Switzerland, and accompanying Swedish citizens have the right to seek and find employment without needing a work permit.
For accompanying partners who are foreign citizens, the same rules apply as for third-country nationals who are married to citizens of these countries. They are thus exempt from the work permit requirement. Under EU Regulation No 883/2004 (entered into force on 1 May 2010) on the coordination of social security systems, you are covered by the social insurance system (and unemployment insurance) in the EU/EEA country in which you work.
Arbetsförmedlingen’s (Swedish Public Employment Service) website has a fact sheet with information about where to find jobs, social insurance, taxes, unemployment insurance, etc. for the specific country you are interested in. Also see the Arbetsförmedlingen/EURES fact sheet with general information/a checklist about jobs in the EU/EEA and Switzerland.
EURES – the European portal for mobility in working life – has information about work and educational opportunities in Europe.