I was always told an illegal alien or anyone visiting from a foreign country (NOT A US Citizen) is NOT subject to jurisprudence of the United States. They are subject to the jurisprudence to the countries they are from. If your on vacation in France and you wife has her baby, it is NOT a French citizen. You are citizens of the United States therefore the child is American.
My daughters were born on English soil while I was stationed in London, and they most certainly have dual citizenship. To say you are not subject to the jurisprudence of a foreign country would mean that you could not be locked up for breaking their rules. Americans overseas are locked up every day! In reality, only diplomats can break rules and not be arrested.
The rub for me (and my daughters) is that since they have dual citizenship, they are not Natural Born Citizens of America, therefore can never run for the office of President. Bummer.