People in the United States are free to be citizens of more than one country. They can even be citizens of more than two. People who are from Mexico, for example, can be citizens of Mexico and citizens of the United States. Dual citizenship can happen when someone is born in the United States to foreign-born parents, for example, so it's not all that uncommon.
I'm a white male, but from what I understand, Latinos and Latinas are from North America, whilst Hispanics encompass Spain and Mexico. The complex overtaking of the Spanish conquistadors over the Chicanos of America is splattered with blood. So this identity should not be decided by me, a Scandinavian Caucasian.
If people who are latinos or hispanics follow the rules of both countries (or even more), and they are eligible for dual citizenship, then they should be allowed to do so, as any other citizens. However, if the question refers to if they should be allowed to keep their original citizenship, then no; if you apply to be a citizen in the US, you have to renounce other citizenship.
Latinos and Hispanics can both be dual citizens. It would make no sense to allow for one group to have a dual citizenship while denying the other. We already allow for dual citizenship's from other countries, so we should be allowing it for all groups in order to be fair.
Anyone in the world can hold dual citizenship if they want to. No one has to renounce citizenry of another country to be a citizen of another. Foreigners are often citizens of the United States and another country all the time. This shouldn't be a problem whatsoever in the melting pot of America.