Those with dual citizenship bear the responsibility of being responsive to two different countries in terms of fulfilling their respective residency requirements and also with respect to paying taxes. Aside from being able to live in more than one country, another benefit of dual citizenship is that you can participate in government programs they both offer.
One type of individual that may quality for dual citizenship is someone that became a naturalized American while still maintaining his or her other citizenship. Although the United States has no particular policy in place that formally recognizes an individual's right to dual citizenship, they don't require him or her to give it up either. So long as the other country allows for dual citizenship, they can keep both.