When using 2 weapons which both have Concussive 1, provided enough advantage is generated (2 for the first Concussive, 2 to activate the 2nd weapon, and finally 2 to activate Concussive on the 2nd weapon, sans Jury Rigged or Imbue Item), do these effects stack causing the target to be concussed for 2 rounds? Or would the 2nd Concussive only be useful when affecting a 2nd target?

If you generate 6 advantages, so enough to activate both concussive qualities and trigger the second weapon, I'd say that they'd stack if you decide to use both weapons on the same target. I'm pretty sure this is commented in the CRB, under the disorient, immobilized and other effects.

EDIT: see page 218 EotE CRB.

Thank you, I missed that part of the description. It explicitly answers my question.