This is more of an economics question than personal finance. That said, I already started writing an answer before I noticed, so here are a few points. I'll leave it open for others to expand the list.
For the "using" country:
- They don't have to invest in expensive currency minting equipment and all the security protocols that go along with owning/using it.
- Bigger currencies (in terms of # of users) tend to experience less volatility than currencies that are not as widely used. This is because market disruptions in one local economy using the currency are tempered by otherwise stable economies in other jurisdictions. Usually, the health of the economy in the issuing country will have the greatest influence on the overall value of its currency, so currencies issued by countries that have large/stable economies may be desirable to smaller nations that have recently experienced a lot of volatility in their market, eventually prompting a switch.
- The local government has less control over the export value of locally produced goods. This is because the central bank of the issuing country can, to a limited extent, increase/decrease demand for their currency by raising/lowering interest rates. This in turn alters the cost of importing/exporting to countries using different currencies.
- If some catastrophe affects the economy of the issuing country and devalues its currency, your own economy will be subject to more expensive imports (though your exports will be cheaper for foreign customers, so this is not always a negative).
- The country cannot "print" more money to pay back debts, and thus may be forced to default, whereas a country whose debt is in its own currency can just "print" more money and never needs to default (albeit it may cause inflation).
For the "owning" country:
- Another country switching to your currency, whether gradual or instantaneous, increases demand for the currency. This makes the currency more valuable, so you can import goods more cheaply (though this will also make your exports more expensive to foreign customers).
- More users of a currency will generally increase its stability. If something terrible happens to the economy in your own country that would normally devalue its currency, the otherwise stable economies in other countries that use your currency may help prop up its value.
The flip-side to the argument that more users means more stability is that the impact of a strong economy (on the value of the currency) is diluted somewhat by all the other users. Indeed, if adopted by another country with similar or greater GDP, that economy could end up becoming the primary driver of the currency's value.
It may be harder to control counterfeiting. Perhaps not in the issuing country itself, but in foreign countries that do not adopt new bills as quickly.