This is politically impossible to stop. Sweden has not closed its borders, so there's no one to stop danes from entering. And Denmark can't close it's borders to danish nationals, so they can't be stopped from returning home.
The Danish Government has made clear statements saying that they strongly urge people to not go to Sweden
Remember, this is the same country that during the Syria crisis stripped incoming refugee of their belonging then dropped them off by the Swedish border.
Edit: since people are downvoting this:
https://www.theguardian.com/world/2016/jan/26/danish-parliam...
https://www.thelocal.se/20150910/danish-police-allow-refugee...
https://www.rt.com/news/445552-denmark-refugees-island-unwan...
Italy, and to a lesser extent Spain, did not.
Same day I wrote my OP it actually rose above Italy with a whopping 2200 new cases at once.
We'll see if all the students celebrating these last few days will make that rise again.
Either way my point was that the spread compared to the number of deaths is the real scandal in Sweden. Low spread, high number of deaths indicates something is wrong in healthcare.
Both Italy and Spain imposed strict quarantine measures in a time where the epidemic was already rampant and killing hundreds of patients a day.
Thanks to that, now they are experiencing zero deaths a day.
Number of deaths is more reliable, though even those numbers can be hidden. For example, in Florida the number of COVID-19 deaths is fairly low, but the number of pneumonia deaths is more than 6 times higher than normal all of a sudden.
NZ locked down while spread was limited, and has had no new cases for 12 days, and only one remaining active case, as a result - but again, it was the timing of the lockdown - if it had happened 2 weeks later, who knows how it'd be doing.
Yet Sweden are seeing a pretty high mortality compared to those countries.
So my conclusion is that quarantine has done no difference to spread but the state of health care is affecting mortality.
And not only Norway is making mistakes in this new experience. The fact that Denmark closes down tourist areas, and the border, but allow their citizens to cross the bridge to Sweden and enjoy all the tourist areas there, and then come back to Denmark, that's just silly.