There are several things at play here.
First is that most countries require visiting boats to have ships and operators licenses. US citizens operating in US waters for recreation are exempt, but as soon as you go somewhere else your host country will likely require you to have a license. If you have the appropriate FCC licenses (ship's station and operators) they are recognized by other countries, just as we recognize theirs. Will you get caught? Who knows, and the risk is yours to judge.
The second thing is the MMSI number. No matter where you get the number, your radio will broadcast it and it will get received. The issue has to do with the databases that contain your info, or not. The international database is maintained by the International Telecommunications Union (ITU), and only MMSIs obtained from the FCC get entered into that database. The boatus MMSIs do not. Here's a link where you can look up your boat or someone elses
Particulars of Ship stations
If you put out a mayday in some faraway place, the responding organization will look up the MMSI in the ITU database. From that they can tell whether they are looking for a white skiff, or a navy blue 80' trawler, along with contact info, etc. If you are not in the database, you won't get ignored, but you are also not giving them the most help that you can, and it's your ass that's in trouble, not theirs, so you should give them as much help as possible.
As has been mentioned, the hardest part about getting the licenses from the FCC is navigating their web site. When you apply for a ships station license, there is a check box where you say you also want an MMSI. They will take your money and give you both. It's painless once you find the link.
So if you think you will ever leave the country, go the FCC route.