The Oath goes:
"I hereby declare, on oath, that I absolutely and entirely renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state or sovereignty, of whom or which I have heretofore been a subject or citizen; that I will support and defend the Cons ution and laws of the United States of America against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same;................."
By taking the oath and becoming an US citizen, I renounced my Argentina and Italian dual-citizenship.
Some countries like Italy allow you to apply for dual citizenship after taking the oath (if you were previously an Italian citizen), so I could do that. Argentina, I think, always treats you as a citizen if you were born in the country, regardless of what you do. But if you identify yourself as an Argentinean national, then the US might be limited to help you in case of trouble.
It's all very complicated, but I'm happy to be a US citizen