I'm going to move to the Netherlands. I have a full-time working contract valid 1 year. I will not be able to join the "Higly skilled migrant" scheme.
Is it possible for me to also work as a freelancer?
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As EU citizen (except if you come from Croatia, for the time being), you are in principle allowed to take on any job you like anywhere in the EU and also to freelance on a par with locals (with a few restrictions for sensitive jobs, e.g. in the military and sometimes some complications to have your credentials recognised if yours is a regulated profession). You do not need a work permit and should be treated like native workers.
In principle, EU countries can ask EU citizens to leave if they don't work and don't have financial resources but that's not an issue for you as you do have a job (see Moving to Germany with fiancée (both EU citizens) because of work reasons. What if she does not find a job?, US Citizen with EU Spouse who doesn't yet have a job and this answer on the politics website for more). If you will, you need a job to have a right to reside but you don't need anything to have a right to work or freelance.