How can I to contact users?
Can I contact duolingo users privately without using another social network? There is a way to find duolingo users, who are new at a language?
I don't have Facebook and if that's the only way… well… I better don't say anymore.
There is no private message feature on Duolingo. You can, however, click on someone's name/icon to go to their profile (or "stream" as Duolingo calls it) and post a message to them there. If you want to talk privately, you'll have to use something else, such as a social media site or email. Be careful giving out any private information though! It's a good idea to set up a separate email (or social media account) to share here, since everything is public.
Is it still possible to contact someone by dropping a message on their profile? I don't see such an option. I joined a club recently, I made some friends there and then suddenly I was kicked out from that club (no idea why). Now I'm able to see my friends profiles but there is no way to contact them anymore.