It's really simple. If the spies were not subject to US jurisdiction, then the FBI could not have arrested them. Instead, they would have to have been declared persona non grata and expelled. But that's not what happened. The cuffs went on, and they appeared in court in prison attire, very much in custody and very subject to US juridicion indeed! Criminals, not diplomats!
So, under the 14th, that means their kids are natural born US citizens, eligible to run, upon attaining the age of 35 (assuming they've been hear 14 years).
The rights and responsibilities are pretty much inherent - but the privileges come only when one fulfills the responsibilities assigned to them. Such as being properly registered with the government in order to claim one's children as U.S. citizens.