When you create a Duolingo account, you choose a username and provide us with an email address, which is not public. It is also possible to sign up using Google or Facebook credentials. The username/name you chose can be seen by other users of Duolingo. Please note that your friends can find you via the search function if they know which email address you used for your account.
If you are a student or teacher using Duolingo for Schools, your email can be seen by your teacher (and students can see the teacher's email).