As Cerberus wrote about 'U' English, replying to "How do you do" with "How do you do" used to be the case among some classes in England (at least), but it seems to be (sadly) nearly extinct. Kate Fox writes in the first chapter of Watching the English (which is about talking about the weather):
We used to have another option, at least for some social situations, but the ‘How do you do?’ greeting (to which the apparently ludicrous correct response is to repeat the question back ‘How do you do?’) is now regarded by many as somewhat archaic, and is no longer the universal standard greeting. The ‘Nice day, isn’t it?’ exchange must, however, be understood in the same light, and not taken literally: ‘How do you do?’ is not a real question about health or well-being, and ‘Nice day, isn’t it?’ is not a real question about the weather.
So in this usage, "How do you do?" wasn't an actual question about the person: it was just a meaningless greeting, and for instance what one might say when introduced to someone (in lieu of "Pleased to meet you"), presumably while tipping one's hat. In reply, the other person, also wishing to make the same meaningless greeting, would say "How do you do".
Note that "How do you do" in this usage was even spoken as a statement, not as a question (i.e., without a rising tone at the end).
These days "How do you do?" is more likely to be interpreted as an actual question. Interestingly, as n0nChun observes, one does sometimes hear a similar exchange these days, with "What's up" getting the reply "What's up", or even just "sup" — "sup".
"I'm fine" is the standard happy medium. "I'm well" marks you as British English,
while "I'm good" marks you as casual American. "I'm good" is incorrect in the traditional sense, but is broadly used to affect careless informality.
"Good" is traditionally a moral qualifier, or an indicator of ripeness, not human wellness as such.
Best Answer
Yes. Depending on where you are in the English-speaking world and on your relationship with the other person, you might reply with any of the following:
Certainly, several other possible permutations. No response at all would be considered rude anywhere. Just saying, "Fine," in response without a "thank you" or a reciprocal question would also be considered brusque, standoffish or rude; or the asker may simply conclude you are in a bad mood. The first response is the norm in places or cases where the respondent does not deem it polite or appropriate to respond in kind with a "How are you?"