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[Do] you envision any trouble by showing her US passport in/out of US and then her Italian passport in/out of Italy but with a different first name[?]
She will book her flight tickets using the name appearing on her US passport for safety.
No. In my experience, Schengen-area immigration officers at airports do not match the passport details of arriving EU citizens against the manifests of recently arrived flights. However, my passports have the same name, so they may be matching the name and date of birth and not asking about the different nationality and other document details. I've also not done this in Italy for well over 10 years, and it's possible that the practice is different in different countries. It's also possible that the practice will change in the future.
Therefore, I would add that there is no problem with your wife showing both passports to the Schengen-area authorities when she arrives. Both countries allow dual citizenship. The US passport proves that she is the person on the passenger manifest (if such proof is needed), and the Italian passport proves that she is an EU citizen. So, even if a requirement to match arriving travelers to a passenger manifest exists or arises in the future, there should be no problem.
Once ETIAS is implemented, it will probably be necessary to show both passports when checking in for flights from the US to the Schengen area. This is because the name on the ticket will match the US passport, but a traveler with a US passport will need ETIAS authorization or some other document exempting the traveler from ETIAS. The other document in this case is an Italian passport. If the airline then puts the Italian passport into the advance passenger information, there will be no need to worry about the different name in the US passport.
If she should ever have an air ticket in her Italian name, the same is true in the US. At check-in, she would need to show the Italian passport to establish her identity as the person named on the ticket, and she would also need to show her US passport so the airline can send that information to the US. On arrival, she would show the US passport, but if any question were to arise concerning her name, she could show the Italian passport, too.