A person who has a dual citizenship is a citizen of both the countries in reference. So he/she has all the rights that a citizen of each of the countries has. And this includes the right to vote as in a democratic process.
As the concept of dual citizenship is not recognized by all the nations and many forbid any foreign citizen from having a citizenship of the country without denouncing their original citizenship, it is not possible for people to be able to vote in all nations. But nations which allow dual citizenship allow the person with the dual nationality to be part of the democratic process just like an ordinary citizen of the nation.
A classic example here is US Governor Arnold Schwarzenegger, who is a governor of a state in the US and is also an Austrian citizen.