That is true. There is no right without duty.
When you claim your rights, it is your duty to ensure that you also give others their right to claim their rights.
For example, you claim your right for freedom of speech. Your right to freedom of speech ends when you stop others from exercising their own right to their freedom of speech.
You can't simply exercise your right when in doing so, you deprive others of their rights. It is your duty to let others claim their rights as you claim yours.
Therefore, there is no right without duty.