Part 121
From your post, it sounds like you're asking about US 121 operators. It really depends on your POI, on what is required. In our neck of the woods, they really don't even consider Mexico foreign operations. I'm talking about the FSDO's not people like Customs.
The only training our PIO requires is overwater, since we do transverse the Gulf and Class II navigation, even though we do operate within the Operational Service Limits of Class I navigation fixes. I have not found any areas within the Gulf and the Carribbean that requires HF communications. I have covered a good portion of it, but not all of it yet.
I have heard that our local office is requiring International Operations Manuals for operators that are getting RVSM approvals at this point in time. I believe it is because the RVSM approvals are for the most part for either North Atlantic or Norpac operations. We'll see when D-RVSM comes about.
I am basing this on my experience under Part 135. However, understand that Part 135 and 121 inspectors operate out of the same book. It is called Form 8400-10. Hope that answered your question.