Not merely a lot (in Sweden, anyway) - I'd say the vast majority of Swedes are fluent in English; of course more or less so but I don't think I've ever met a Swede who couldn't at the very least hold a basic conversation in English.
And that goes for authorities too. Had a non-Swedish partner for a few years who always communicated in English in all interactions with Sweden's governmental services - no problem. All important government websites are available in English. Knowing Swedish isn't even a requirement for citizenship.
So while knowing Swedish is certainly very beneficial and recommended, you can certainly get by without it, and work for many interesting companies (at least as a developer).
My wife and I spent two weeks in Sweden (well, two of those days were spent in Oslo...), and we only encountered one person who didn't speak English. He was a pizzaman/bartender in Nynäshamn, and luckily all the dudes sitting at the bar translated for us in drunken semi-unison :)
(Honestly, though, we probably would've gotten along fine without the translators, as we'd spent a couple months with Rosetta Stone before traveling... we were/are far from fluent, sure, but good enough to order a pizza, I'd imagine ;) ).
I lived in Denmark for 5 years and yes, it has been like that.
However, all the paperwork you will need to handle will still come in the official language, not in English. All contracts and legal documents will be in that language as well. So not knowing at least the basics will handicap you severely.
I have learned enough of Danish to be able to read the paperwork and do some basic writen communication. That's really the minimum to get you by.
Spoken Danish was a bit too much - the saying that Danish is not a language but a throat disease is not far from the truth!
I'm in Norway, and it is similar here. Not all industries are so kind to non-Norwegian speaking applicants, but government and everyday life usually isn't an issue. Half of the official paperwork has English as well as Norwegian - and English transcripts and official documents are OK to turn into the government without further translation.
That said, as soon as I started getting some actual Norwegian language skills, it improved.