Your teacher gives you the code. In a class, your teacher sees your name, which assignments you finish and how you do on the decks of the class, including which quiz questions you got right. They do not see your friends, messages or other decks.
Your classmates see your name, picture and XP on the class ranks. You can hide your name from your teacher and your classmates, or leave, whenever you like, in the Privacy tab of the class.