Whenever you are matched for a position, the Employer who posted the position cannot see anything but your Anonymous Profile. To phrase it another way, your identifying information is masked by default.
While you're masked, the Employer can chat with you and review your profile. If the Employer wishes to access your full profile, perhaps to set up an interview, they will request permission to "unmask" you.
If you agree to be unmasked, the Employer will be able to see your full profile. This includes your full education and work history, name, and portrait. If you aren't interested in the position, or for whatever reason do not wish to be unmasked, simply deny the request.