The only real responsibility that the Declaration of Independence places on people is the responsibility of fighting back if their government is not protecting their rights.
The Declaration of Independence begins by talking about the duties of government. It says that all people have the inalienable rights to life, liberty, and the pursuit of happiness. It says that the only reason for governments to exist is to protect these rights. So far, this does not place any responsibilities on the people. Instead, it simply says what government is supposed to do.
But what happens if government does not do what it is supposed to? This is where the people come in. The Declaration says that people would then have the right to “alter or abolish” their government and set up a new and better one. It goes on to say that, when the government is clearly abusing its power and taking away the people’s rights,
it is their right, it is their duty, to throw off such Government, and to provide new Guards for their future security.
From this, we can see that the Declaration says that the people have the responsibility (or the duty) to overthrow their government and create a new one if the government is trying to take away their rights.