Hmmm, so similar to the comments below, how would you differentiate sides of the document, such as the essay must be at least 4 double-sided pages long, or the notes may be no more than 2 sides of A4 - something you could easily hear in a college or university, for example- rather than edges as in the amusing, octagonal example Helen gave?
I feel like "page" always refers to something with written text on it, unless you specifically say "blank page". But I'd probably "can you get me that piece of paper?" if I was asking for a piece of paper whereas I would say "you're missing a page" if you printed something out that was (for example) 5 pages, and only 4 were printed. To me, sheet/piece of paper refers to the actual paper while "page" is a little more abstract in it's meaning. It suspect they may actually be similar in swedish.
I wish the people who share their opinion about this and similar things would also share what their native language is and where they are from. I think I know Berniebud is a native speaker of American English (and HelenCarlsson is Swedish of course), but for the rest of you, I have no idea. Is sides long maybe a British thing? To be honest, we have a lot of users who are not native speakers of English but may have very strong opinions about English usage anyway, and I'm not always sure their intuitions about English are better than anyone else's.
Sorry to not have said earlier. I'm a native international English speaker (international school, but mostly British influenced) and I'm a teacher in a school in London. "Sides (of A4)" is definitely used, particularly to avoid confusion and when describing written or printed documents. It might very well be just in academic/school situations though, but I feel it's pervasive enough for everyone to use it generally. If someone says "3 pages" it could mean six sides or three sides. Using "sides" is unambiguous.
Native English speaker (Australia). I have heard the term 'sides' only in an academic context: where the teacher/lecturer was making very clear that they want to have to mark no more than 4 pages of text, they would sometimes say "four sides of A4 in 12-point type" or something like that, for the avoidance of any (almost certainly knowing & deliberate) misinterpretation that might lead a student to submit an 8-page paper. Basically I think it is not at all common, but sometimes when the speaker really wants to rule out any ambiguity about the length of a document, they might say "sides". I don't expect I would ever hear anyone say "I've just read a 200-side novel."