I believe that I would give out my last name if asked. I've had employment where we regularly received death threats and I still felt it was the right of my clients to have my last name. Now. Just TRY to find me using my last name. Not as easy as you might think.
In practice though, my last name is a mouthful to say and nobody can spell it correctly. So as an everyday way of introduction it wouldn't be terribly helpful (My first name is difficult enough for some people, I go by a shortened version of it for consumer ease).
Also, in the myriad times I've been hospitalized, my nurses have always both stated and written their names (most usually the first name, I believe) on a whiteboard in my room. I have to confess that at the time I was so ill that I really didn't pay much attention to anyone's name. Sometimes, I barely even knew my own. I wouldn't have remembered anyone's name after leaving the hospital.
I remember the faces and the kindness of many of the nurses and thanked them at the time when I could (it's amazing to me that nurses can be so difficult with each other considering how well they took care of me and how kind they were btw - just as an aside).