Learn English – Is it right to say “Thank you” in the response of “Thank you”

In common conversation in the US Midwest I rarely hear "Not at all" or "Don't mention it." "No problem" is very common, and "You're welcome" is also pretty well-used.

My personal usage:

I use "Not at all," "Don't mention it," and "No problem" when the activity I'm being thanked for was really no big deal. I use "My pleasure" when emphasizing that I'm happy to be of assistance (often in a customer service context), and "You're welcome" when the action prompting the thanks was a little bit of a bother. In essence I use different phrases to indicate how "thanks-worthy" the activity was.

That's probably not common usage, though.

EDIT

I think I misrepresented what I originally meant, so here's a little clarification.

If someone thanks me for something I always do (for instance I always cook dinner in our house) then I would say "No problem" or "My pleasure" depending on context. If I did a chore that was someone else's responsibility, I would say "You're welcome" even if I was happy to have done it, because it took an extra effort on my part, not because it was a "bother."

You're right that "thank you" could be used in either a singular or plural setting. So "thank you both" and "thank you, Raj" both work.

But "thank you" is a phrase, a shortened form of "I thank you" (http://www.etymonline.com/index.php?term=thank+you); to split it and try to use its component parts as if they were joined -- and as if they were both nouns -- is misguided. As soon as you split "thank" from "you", the words become separate entities.

"Thank" is a verb, which is why, as you suggest, it demands a subject ("I" in the first person) when not used as a directive (e.g. Thank your mother). That's one reason why "Thank to both of you" doesn't work (i.e. is grammatically incorrect), and why "I thank both of you" does.

"Thanks" is a noun, so "Thanks to both of you" would be another grammatically correct option here.