In a standard textbook introduction between two people, they normally shake hands (a strange social custom in itself when you think about it) and exchange names. This type of interaction has gone on pretty much in this same fashion for hundreds of years. For me and most others, we do our best to remember the other person's name and its proper pronunciation in later interactions. This is a form of respect and caring. I would certainly correct someone if they mispronounced my name, or even worse, called me by the wrong name. I would expect that others would tell me if I did the same.
Being in the field of academics for so long, I have come across many different races and cultures. It is part of the melting pot of this environment. Chinese, Japanese, Russians, Europeans, Indians, Pakistanis, Spanish. It is curious to me how quickly people throw their names away when they come to the U.S.. They feel their names are too difficult for us to pronounce, so they adopt a common name like "Bob" or "Susan" immediately upon deplaning at one of our major international airports. For me, I would not do this. My name is part of who I am and represents my identity. It is clear though that others do not put as much importance as I do into what others call them.
The impetus for this posting came about from two very funny observances of this sort that I came across recently at work. One was with an Indian student named Biplab and the other was with a French woman named Fatiha. Both of these folks were visiting my laboratory to give talks on their research. At the end of such talks, it is standard for the speakers to answer questions for 10 or 15 minutes. At the end of Biplab's presentation, everyone kept referring to him as "BeeBop" (and in one case I even heard "Skee-Boop"). Similarly, at the end of Fatiha's talk, everyone kept referring to her as "Fahita" (well, it was just before lunch time). In all of this, neither Biplab nor Fatiha's demeanor or expression changed. Perhaps they have gotten used to their names being butchered on a regular basis. After all, we American's all look and sound the same anyway.