Tight-lipped danes are want to say that the enormous friendliness of Americans does not run very deep, whereas the danes underneath their reserved exterior are really very nice people once you get to know them.
The claim is of course bogus. It's just that friendliness is part of American idiom ('Hi, how are you') and not in Danish. Clearly use of idiomatic phrases should not be taken to be actual intrusion on your privacy, nor should they be interpreted as genuine concern. That doesn't mean that there aren't phrases that mean the same thing to an American that 'how are you' means to a Dane.

This is all obvious and well-known. Traveling in America, it takes a day or two to get your own level of friendliness and approachability up to regional standards, but even us tight-lipped people do get it eventually. At least if you're just the very least culturally sensitive.
The more interesting question is how to actually be as friendly as an american, once you're back in Denmark. You can't do a direct translation of american phrases. People would basically try to avoid you if you did, and it is not really obvious how to be as friendly in Danish, except of course that you can smile and talk a lot.