Some parents with European citizenship sometimes do not file an American birth certificate when registering their child for a European birth certificate. I.e. if the country facilitates registration of child based on parent citizenship and not birth location of child, they just never declare the child was born in the US. "It was a home birth"
At international schools in some countries this is an open secret and happens quite often, as long as the parents are not feds and are not under heavy scrutiny. So the kid has two different BC & passports with two different birthplaces. Some even have different birth-dates and names.
Sure saves the poor kids the nightmare of not being able to open a bank account abroad, as a US citizen.
The moral is, plan your money, plan your kids, the federales will never inform on what is in your best interest. Both kids and money come with paperwork attached. Best read up now. And yes, a timely consultation with a lawyer before the fact is most often cheaper than the cost of renouncing US citizenship and the time you waste on declaring while not living in the US.