There isn't any "have to" formula for your reply. Personally, I don't say either of these. I usually respond, "I'm pleased to meet you!" (I might use "glad" instead of "pleased" or, if I've heard a lot about this person and am especially happy about meeting him or her, I might use "delighted".)
Thanks for your comments. But I'm sorry I didn't explaine well my question. I'm asking particularly about answering with : me too or you too.
By saying "me too" do you give the impression that you are saying : I am glad to meet me too? I mean when you say: me too, does it mean you have taken out the whole phrase but these two words? And like this you are actually repeating the other's person's phrase but with "me too" ending?