Sorry @LeonidS , I had a misspelling. "I want visitors of my profile page to NOT be able to see that navigation". It should only be visible to myself as the owner of that profile.

Its more meant like: I want to have links visible to myself to edit my profile (like changing the email address), but visitors of my profile shouldn't be able to see those (as they shouldn't be able to edit my email address).

Thanks @Ivan Mishchenko, but neither of them worked as described above.