(1) Dollarization is not pegging to the Dollar (which is what likearock refers to, a currency peg.), rather adopting the Dollar as the currency of the land. No other currency is circulating as legal tender.
Other regimes are found in Argentina which uses a currency board to maintain a fixed relationship between their currency and the dollar, however the dollar is not formally speaking legal tender, although it is de facto used. Technically, however, official and on-the-books transactions are in the Argentinian currency.
For the US trivial ones already faced due to large dollar holdings outside the USA anyway. Our official policy is that we make no allowances for outside needs. Problem of counterfeiting is perhaps extended to small bills, and even coins per what appears to be developing in Ecuador.
For Dollarized country:
(a) You lose monetary policy, i.e. the ability to use the money supply to effect short term adjustments in the economy as well as to respond to liquidity problems. Essentially your money supply is at the whim of the Fed. Clearly this takes away an important tool in managing an economy. Of theoretical value for Ecuador now, but in the future… If their economy is not linked to the US --i.e. if they need more liquidity when we need less, they could face serious problems. Also, if their trading partners currencies are floating against the dollar, and the home economy trades significantly with them, there could be serious impacts on trade patterns. Given the dollar’s value, probably encouraging imports and discouraging exports.
(b) Practical Application. In countries with high illiteracy rates our currency, not terribly friendly to the illiterate given its all the same size and color, may present serious transaction costs. People literally not knowing how much bills are worth. This will decline, but nonetheless imposes an added burden on the populace which may not be strictly necessary.
© Externalizes policy, probably retarding the development of local political discipline and thus the development of healthy local public financial institutions/traditions.
In short, I think its rather a bad idea for the adopting country. Stunningly bad really. The same results can be achieved different ways, and with better (in theory) long term consequences.
But we will see, Ecuador and El Salvador will be out guinea pigs.