An updated answer for future readers:
Private individuals cannot seek insurance through Kammarkollegiet, it seems. Organizations such as universities can, and if you are a tuition-paying student will probably do so for you.
In my case, I was able to acquire a personnummer smoothly with only my Finnish passport thanks to the Inter-Nordic agreement.
If you are an expat from the EU and have been residing in the U.S. (or anywhere else, I suppose) for the past few years, you're in a series of catch-22s regarding personnummer, health insurance, employment, and a bunch of other things. I've found multiple articles and posts around the Internet of people in that boat, and the best solution seems to be to get a taxable income in Sweden so that you'll "automatically" be registered with skatteverket.
Old answer below:
Alright, we contacted Your Europe Advice for help. Here's the short version of what we were told. Keep in mind that this is for our specific case, and may not apply to everyone in a similar situation:
In Sweden a registration with the tax board is necessary to obtain a personal number. Swedish personal number identity is the starting point for a variety of services and registrations in the Swedish society. Without access to this personal number (pin number) civil society life is very difficult in Sweden. The difficulties of getting the personal number is often linked to the registration with the population (census) and are often linked to the requirement of being resident for at least one year in Sweden and being fully covered by an insurance (1).
In order to be protected by the EU social security and be issued an European Health Insurance Card (EHIC) under the EU legislation (provided by Regulations 883/2004 and 987/2009 coordinating the social security within the EU) it is not a question of nationality or citizenship. You are protected by the social security of one EU country if you have been registered in a national social security system (2).
- As I have been living outside the EU...
- I have therefore not not been insured under the EU social system in Finland
- I cannot obtain an EHIC from Finland
- I must be protected while I am living, traveling, and studying in the EU/Sweden in order to obtain a personal number (personnummer)
We recommend therefore that you take an insurance with the "Kammarkollegiet" in Sweden. These insurances for students give not only a protection in Sweden but also when you travel in the EU. You will find these insurances on: