The Limits Of Monetary Policy Independence
First of all, it is not necessarily the case that independent monetary policy will really be sounder than the policies of some other central bank. Particularly in many third world countries, adopting the policies of some sounder Western central bank is probably better.
And secondly, particularly for smaller countries, monetary independence is partially or entirely illusory.
The best example of this was the case of Iceland, that raised interest rates really high to contain its speculative boom. Yet that only helped push the Icelandic krona to increasingly overvalued levels, increasing the current account deficit and borrowing in foreign currencies.
Iceland is of course a really tiny country (the population is only about 300,000 or 1,000 times smaller than that of the United States or the Euro area) with high dependence on foreign trade, so for larger economies, national monetary policy will usually not be quite as ineffective as in Iceland. Generally speaking, national monetary policy will be more effective the bigger the country is. Yet even for larger countries the exchange rate movements caused by interest rate changes will constrain the effectiveness of attempts to contain bubbles and imbalances.
One example of this is the effects of the Fed's "quantitative easing". This has caused the U.S. dollar to dive against just about all currencies. Given this, other central banks face the choice of either easing too so as to limit the appreciation of their currencies, or accept a really dramatic appreciation of their currency. Regardless of what they choose, the Fed's policy will increase imbalances and bubble tendencies in their economy. Thus, even if they have a floating exchange rate, other countries will feel the effects of the Fed's policies, meaning that their monetary policy independence is more limited than many people think.