I don't think the US will notify another country. However, the other country will likely ask you about your status in the US when you renew the passport, or might ask you if you have acquired any other citizenship. In that case you would either have to lie in the application or find some creative explanation if you renew at a consulate and they demand to see a work visa or green card to show you are in status. I know people have been able to pull this kind of trick in the past depending on the country. In my view every country should allow dual citizenship.