EU citizen without EHIC

Hello everybody!

I am a dual Chilean/Italian citizen considering undertaking postgraduate studies in Europe (Denmark and Sweden are my options) starting next year. In both cases I would apply to universities as an EU national, so I won't need a visa. However, I am concerned about healthcare. Since I live in Chile, I am not entitled to an EHIC card (Italy only grants them to citizens who reside in the country), thus I won't be able to register as an European citizen for healthcare purposes neither in Denmark nor Sweden. What can I do in this case? Would I be able to register as an non-EU/EEA national for healthcare purposes even if I don't have a visa (because I don't need one)?

I hope you can shed some light on this! Thank you very much in advance!

EHIC or CSI (comprehensive sickness insurance i.e private medical insurance) usually need to be possessed by the EU national students in order to excercise its treaty rights as a student where they are supporting their non-eu family member or willing to seek PR of that particular EU member state usually after five years. Otherwise you won't need of it as you can still enter and live freely in any EU member state.