I was about to post the same question. I see that yours was asked a year ago...have you since discovered a difference? (Interestingly, I received this sentence previously as a translation exercise and I had indeed written "are you known in this city" and it was accepted. Then, I got the same sentence as a multiple choice question and I see now that it has the other meaning- I too would like to know the difference, if any.)

My best guess is "are you knowledgeable in this city" and the meaning is more implied than stated.

'den'(masculine/feminine) and 'det'(neuter) mean 'that' or 'it'.

'denne'(masculine/feminine) and 'dette'(neuter) mean 'this'.