From what I’ve read, it’s the responsibility of the SOS of whatever state you first file to run for office in to determine your eligibility. This would be Illinois, in Obama’s Chicago case.
The final arbiter seems to be the Solicitor General. Ted Olson (widower of FReeper Barbara Olson, who was killed on 9/11, who called her husband from her cell phone on the plane describing what was happening) was the analyst on Claire McCaskill’s Senate Resolution to decide McCain’s eligibility for office, the one into Obama tried to inject some language helpful to his case back in April.