How does this work? Does Ecuador just pay the US mint to manufacture extra dollars so they can use them?
Anyone know how that works?
There’s more than enough US currency in circulation for Ecuador to get some without making special requests.
Ecuadorean banks can request currency from the Federal Reserve system the same way any other bank can.
Panama has used the US dollar as its currency throughout its history. It’s never printed its own bills for general circulation. (It does have its own coins.) Historically there was always enough coming in from the US bases and the canal.
El Salvador also has a dollarized economy.
Panamanian banknotes, denominated in balboas, were printed in 1941 by President Arnulfo Arias. They were recalled several days later, giving them the name “The Seven Day Dollar.” The notes were burned after the seven days but some one balboa notes can be found with some collectors. These were the only banknotes issued by Panama and U.S. notes have circulated both before and since.
Yeah, but hang on - what would they use to pay for it? More dollars? 
Yes, but not dollar bills – most money is not in note or coin form, but is in the form of bank accounts, etc.