Mixture of both, I think. The fact that Danes speak quite good English, and don't expect foreigners to speak Danish, means that they switch to English as soon as they realize you aren't Danish, so it's quite easy to get by in English. But I've spent some time attempting to study the sounds, and I just cannot make most of them in a way that Danes can understand at all (e.g. saying place names, even those I've practiced!). To a somewhat lesser extent, I also have trouble parsing them, though I can recognize some stock phrases that I've heard often. Oddly enough, the Swedes I know can't actually communicate with Danes in spoken language either, despite written Danish and Swedish being nearly identical. They will typically switch to English to talk to each other, even though officially they're supposed to be able to communicate (Danes<->Norwegians and Swedes<->Norwegians seem to have more luck).
> Oddly enough, the Swedes I know can't actually communicate with Danes in spoken language either, despite written Danish and Swedish being nearly identical.
It's not that odd. Take threedaymonk's example: English is written almost the same way in New York and in Glasgow, yet an American in Glasgow might have some difficulty understanding people at first. And of course mutual intelligibility between Danish and Swedish will be lower than between New Yorker and Glaswegian.
Written Portuguese and Spanish are also very similar, and while it's possible to keep a "bilingual" conversation with some effort, I (a Portuguese speaker) and my Spanish-speaking friends just use English instead.
Native Swedish and Danish speaker here. I can confirm that Swedish people usually do not understand Danish, whereas Danes have an easier time with Swedish. I think Danish is just plain harder to understand, and I sometimes speculate that my early exposure to it has kind of hypertrophied my general language ability (I used to win awards for English vocabulary knowledge and that sort of thing in high school).
