As a teacher, your name and a link to your profile page will show up in a Google search, but how much information this page display is up to you. You can edit this information by going to the "Settings Page" and clicking on "Privacy".
If your privacy settings are set to "Only My Connections", then the users that you are connected with will be able the only ones to view your name and profile picture.
If this feature is set to "All Teachers On Edmodo", then your profile will be public.
To learn more about your privacy settings, please visit this page.