Here is a good and simple explanation of this tricky grammar point: http://www.oxforddictionaries.com/words/personal-pronouns

The only time I can think of when native speakers might mix "I" and "me" is when with "I/me" is next to "and" e.g. "Me and my friends are.." or "...with my friends and I" (instead of "My friends and I are..." and "...with me and my friends" respectively). I think this happens because it doesn't strike us as obviously wrong when the pronoun isn't directly next to a verb or preposition. This is grammatically incorrect, but it's still a common mistake among native speakers. Otherwise, I can't think of any situation where someone would say "me" instead of "I" or vice-versa. No native speaker would ever say "Me love..."'; although they might erroneously say "Me and ___ love...", for the reason I described above.

of course not.

The only time a native speaker would say something like 'Me love you' is if we are deliberately using bad or very childish English, as a joke or to create a particular atmosphere. Your suggestion about this being more 'intimate' than standard English may well be correct. Some couples do use 'baby talk' when being affectionate to each other, in order to create a kind of shared private space. It wouldn't surprise me to hear a boyfriend say this to his girlfriend or vice versa.

While it is definitely not grammatically correct and I agree with Jmat's explanation below, I actually have heard other native English speakers say things like "Me love you." They are saying it incorrectly on purpose. It is usually high school-aged girls who are trying to be cute when talking amongst themselves, at least in my experience :) This is maybe what the person who was talking to you was referring to. I wouldn't recommend anyone try it, though, as you risk sounding very strange!