Duome.eu (formerly duolingo.eu) is an unofficial enthusiast project based on Duolingo public data. "Duolingo" trademark, graphics and logos used in this website do indeed belong to duolingo.com and its respective owners.
Should duolingo.com ever decide to shut duolingo.eu down, then so be it. I do believe though that duolingo.eu falls under the fair use concept and everything I do - I do with a great deal of respect.
Update: I actually contacted Duolingo bringing my work to their attention, giving them every detail they might possibly need, so they do have my contacts, my real name, my private email, etc., and I remain open to conversation and any ideas on how can we possibly/hopefully make something nice together. That said, there's always a possibility that there will be no duolingo.eu tomorrow, so if you do enjoy it - enjoy it while you can, and if I don't have another chance to say this: gracias para todos y por todo
Update2: I have been talking to Severin Hacker, the co-founder and CTO of Duolingo. To make a long story short: they do like the idea, but they need to own the domain so duolingo.eu is now redirected to duolingo.com while we can still enjoy the service under the new domain name — duome.eu
Thank you Severin and the entire Duolingo Team!