Which sentence is correct?

OPTION #1
- 谢谢
- 不谢

OPTION #2
- 谢谢
- 不用谢

Both OK in Chinese. It depends on whether the speaker like to use which one. I don't think it's rude to say "不谢", it's more like to say "不用谢" for short. You could feel free to say "不谢", it's definitely a polite way to say so. "不客气" is more common used to answer "谢谢". You could also answer “客气”, this time, "客气" is to say "不客气" for short.  

Which sentence is correct?

OPTION #1
- 谢谢
- 不谢

OPTION #2
- 谢谢
- 不用谢

Generally, option 1 is rude as it is too direct and blunt.  This is because as if the one who said "no need" feels that himself is superior than the one who is thanking him or that it is being said in a sarcastic manner.  Option 2 is more courteous and therefore preferable.