There is a lot of information about getting pre-paid SIM cards in both, Sweden and Denmark; and there are reports about using that pre-paid SIM card in both countries because EU allows network roaming across the zone.
After reading the posts on the best pre-paid SIM card in Sweden, on a recent trip to Sweden and Denmark, I got the Telenor Pre-paid SIM card in Stockholm for about US$30 that gives 10GB of data and 100 calls with a validity of 30 days.
Wanting to make sure this will work in the next country (Denmark), I was told by Telenor store staff (in Stockholm) that it will work in Denmark, which was my next destination. But after landing in Denmark, the phone refused to connect to Telenor DK network!
I visited the Telenor store in Copenhagen, and I was told that for a pre-paid SIM card to work on their Telenor Network, I needed a “Denmark CPR card”, which is essentially a personal registration number. Without that I won’t be able to activate my Swedish Telenor SIM card on the Danish Telenor network!
Being a tourist, there was no way for me to work around that, so I had to spend extra and get a Lycamobile SIM card, which is the only one available without Danish CPR card.
TL:DR – Not all pre-paid SIM Cards bought in one country of EU will work in another country. So make sure to check the availability of network across EU if you plan to get the pre-paid SIM card from one country, to use it in another country.