Nobody 'sets the DC' in an opposed check.
If you make a skill check that equals the DC, you succeed. If you make a skill check that equals the opposed check, the character with the higher modifier wins; in that's still a tie, you both roll again.
Let's say I have a Forgery modifier of +10, and you have a Forgery modifier of +5, and I forge a document. I roll a 10, for a total check result of 20. You roll a 15, for a total check result of 20, when you examine the document. If my check had set the DC for you, you would successfully detect it as a forgery. But instead, since our opposed checks are tied, we compare modifiers; my +10 beats your +5, so you are fooled.
Neither of us 'set the DC' for the other.
-Hyp.