Well, this applies for Cypriot Turkish and Cypriot Greek, actually. If I learn standard Turkish and Greek (I assume those are which are taught on Duolingo (or, will be :)) how would it work out if I were to go to Cyprus?
Are the differences similar to Latin American Spanish and Castilian. Which again, is very vague as different areas within "Latin America" would speak slightly differently as compared to another place within Latin America. And the same would be true for Spain - especially, I presume near France and Andorra; or even Valencia or Barcelona where I believe, Catalan is also spoken (please correct me if I have erred.)
Of course, if I were to start living there for the next 7 years or so, I would have caught the dialect easily - for instance, I prefer Castilian as I was on friendlier terms with my teacher who was from Valencia as compared to the ones from Nicaragua or Mexico. Personally, it just sounds a bit "odd," but I don't find anything too different. It could be that I've just gotten used to the dialects.
So, anyway, what would it be like to know Standard Turkish or Greek and have to communicate with people who speak a different dialect. And from experience, I can't understand my Grandmother who speaks a different dialect of Hindi (despite my Hindi not being fluent and only spoken by my father -- and that my Grandmother barely has teeth left -_-)
As a Turkey Turkish living in Cyprus, I can say that Cypriot Turks seem to have no problem understanding Turkey Turkish. And even though it does sound different to me, I usually do not have any problems understanding the Cypriots, except the few times I was talking to old Cypriots with very authentic accents, probably not even the standard Cypriot Turkish but a regional variation of it. As cometslegend mentioned, probably they are quite exposed to Turkey Turkish through media, and so they are much more familiar with Turkey Turkish than the other way around. In my first weeks here I had a few moments I needed a few seconds to understand what the other person meant. For example, a Cypriot Turkish may ask you "Tamamsın?" whose direct translation into English is either "Are you complete/whole?" or "Are you ok?" which makes perfect sense... in English, but in Turkey Turkish it is never used like that. First time I had that question I was indeed puzzled :) But that is pretty much it, you easily get the hang of it. That being said, the differences in the intonation and wording might cause some slight misunderstandings from time to time. For instance the Cypriot Turks do not use yes-no questions. They just say the positive sentence in a particular intonation to make it into a question. This, to me at first, sounded as if they meant to say the opposite of what they are saying rather than asking a question. Example: A Cypriot says "You would like to drink coffee?!!!" I thought she/he meant "You do not want to drink coffee, do you?" which led me to say "No, thanks, I am fine." thinking they did not really want me to drink coffee. :) This was obviously my false perception and in time I managed to learn that they were simply asking a yes/no question with that intonation.
Well I can't answer the Turkish side of things for you but in terms of Cypriot Greek, the language itself is distinct enough to be pretty difficult to understand for someone who doesn't have any exposure. Its not so much the vocabulary, but the way it is spoken, and more of the culture of the language if that makes any sense. i.e. some Muslim/Arabic sayings make their way into common vocab, due to its geographical location.
Fortunately, given how the culture is, effects of TV and media, religion, business relations etc. most Cypriots can adapt themselves easier to a Greek person than vice-versa.
I read somewhere that about 85-90% of the vocabulary is the same, and the majority of differences come from Turkish, so I guess if you are learning Turkish that would be even easier.