They are required to obey the laws of both countries. Either country has the right to enforce its laws, particularly if the person later travels there. Most U.S. citizens, including dual nationals, must use a U.S. passport to enter and leave the United States. Dual nationals may also be required by the foreign country to use its passport to enter and leave that country. Use of the foreign passport does not endanger U.S. citizenship.
Most countries permit a person to renounce or otherwise lose citizenship.
Information on losing foreign citizenship can be obtained from the foreign country's embassy and consulates in the United States. Americans can renounce U.S. citizenship in the proper form at U.S. embassies and consulates abroad.