I don’t know about the specifics of Ecuador or Panama, but I have heard of countries other than the US being on a “US dollar standard”. If I’m not mistaken, Argentina was on such a standard until a few years ago.
Being on a US dollar standard means that the local currency is exchangeable for US dollars at a fixed rate (not necessarily 1 to 1). So just as under a gold standard, a holder of the currency would be able to demand gold for it, under a US dollar standard, a holder of the currency would be able to demand US dollars for it.
The idea is that the local currency will have value by virtue of its being able to be exchanged for something else of value. This may sound strange, since after all, today’s US dollar is not backed by anything, but in much of the world, the US dollar has much more stable value than any local currencies.
Note that the foreign government (or other entity that decided to issue currency backed by dollars) does not actually print or mint US currency. (I’d be very surprised to find out I was mistaken on this point.) What it prints instead are units of its own currency, which it’s then obligated to redeem for US currency. So it creates no obligation on the part of the US government, or indeed on the part of any holders of US currency.
For example, if I am owed a debt in US currency today and somebody offered me Ecuadoran money as payment, claiming that it’s “just as good”, I could refuse to accept it. (Presumably the Ecuadoran government could force Ecuadoran citizens to accept it, but that’s another issue.)
Using US dollars as a “reserve currency” like this creates more demand for them, so if anything, it would tend to make the dollar more valuable. And I don’t believe the US government could do anything to stop it even if they wanted to. After all, if US dollars are owned by foreigners, it’s up to the owners of those dollars to use them as they see fit.
Note that if a government goes on a US dollar standard, it is then limited in how much currency it can issue, since if it issues too much, it won’t have the dollars to pay out when holders of the local currency demand them. So it’s a brake on local inflation, in a somewhat lesser way than the gold standard was. It definitely makes the value of money less subject to the whim of politicians.
However, there’s always the question of whether the government will stay on the dollar standard. That is, they could at some point in the future decide by fiat that they’ll no longer pay out dollars. Or that they won’t pay out as many. Devaluation, in other words. I believe this is what Argentina did a few years ago, with the result that holders of Argentine currency were not given the US dollars they’d been promised. (Exactly analogous to the Devaluation of the US dollar in the 1930’s: FDR decreed that holders of dollars would not receive as much gold as they’d been promised. The difference is that in one case, the US dollar is the entity backing up something else, and in the other case, the US dollar is the entity being itself backed up.)