There are two ways to “dollarize” the economy.
(1) Black Market
This approach is what’s done in Russia, where the local currency is so soft (read: worthless) that transactions actually happen in U.S. dollars. The U.S. government doesn’t “ship” dollars over there, the free market system brings them. This was actually true during the Communist regime, as well; the ruble was worthless outside the country, and travellers coming in would smuggle in dollars (illegally) for use in the black-market (underground) economy.
This approach actually uses dollars, real live U.S. currency trades hands.
I was in Brazil during the zillion-percent a year inflation. I went into a shop to buy a present for my wife, found what I liked. The shopkeeper said to me, “The cost is Thus, but I’ll give you a 10% discount if you pay in cash rather than in credit card.” (This was because the credit card time delay in 1000% inflation was substantial loss to the store.) The she went on, “And I’ll give you another 10% discount if you pay in dollars.”
That’s the black-market dollar at work. There is an actual currency, but it’s pretty much useless, and so people accept real, honest-to-god dollars.
OK?
(2) Formal Government Action
The second way that a country pegs itself to the dollar is the example of Argentina or Hong Kong. They have a local currency (Argentine pesos, Hong Kong Dollars, etc) but the value of currency is tied to the value of the U.S. dollar. The Argentine peso has been worth about USD 1.00 since the peso was brought into existence. The Hong Kong dollar is fixed at about HKD 7.75 to USD 1.00 for at least fifteen years that I can remember.
Normally, currencies vary in price against each other, depending on a variety of factors (including governments setting the values arbitrarily.) When a country chooses to link its currency to the dollar, that currency doesn’t float, doesn’t vary, remains (reasonably) fixed in value against the dollar.
When a country fixes its currency against the dollar, the local currency is still used; it’s not as if normal people buy things for U.S. dollars. It’s just that the exchange value of the local currency against the dollar is fixed.
Most of continental Europe has similarly fixed their currencies against the Euro, for example.
I presume this is what Ecuador is planning to do, to fix their new currency (currently the sucre, at about 20,000 to the dollar as of January 1) to the U.S. dollar.
Hope that helps.