Only a qualified mechanic can tell you where to spot the damage.
The bumper might cost just a few hundreds, but the structural bar,
beam or whatever could cost you over a grand.
If both parties choose to acknowledge it as no damage, then so be
it. But rather, bring it to a workshop and a mechanic can help you
spot the faults and evaluate what needs to be done. Remember, the
car's resale value is dependant on its state. 