I'll be going back to the US in February for 10 days or so and up until this year, I've always traveled with my US passport and my Italian ID card to prove my EU citizenship since I live in the EU. This year I decided to get an Italian passport as well since Dutch customs were being problematic about my Italian ID card each time upon re-entry. Id I travel to the US on my US passport and travel back to the Netherlands on my Italian one, am I required to get an ESTA on my Italian passport for the trip back or is that not necessary?
US citizens must declare their citizenship when entering the US; and they are never eligible for ESTA (nor would they ever need one).
So you must always use your US passport when you enter the US - and can use whichever one you want when leaving.
[...] If you are a citizen of the U.S., and also of a VWP country, you should not be applying for ESTA. One of the requirements of being a naturalized U.S. citizen is that you apply for, and use, a U.S. passport for your travels. [...]
Simple rule for dual nationals:
Present the passport of each nationality to immigration officials of that same nationality - whether entering or leaving the country. For example, when you’re in the USA, show your USA passport to border officers; and when you’re in Italy, show your Italian passport.
This establishes to them that you’re one of their own citizens, therefore there are no restrictions on your immigration status.
When dealing with airline staff, present the passport of the country you’re travelling to.
This confirms to the airline that you have permission to enter your destination country.
Edit: And to nail down the point to answer the question: no. You’re a USA citizen. You don’t require an ESTA, because you’re one of their own citizens... as demonstrated by your USA passport, which you will present to USA border officers on entry.
For exit, it’s a moot point anyway.