A competing right describes a situation in which by giving rights and liberties to a group of people might be harmful to the rights of other people.
This is something that is often studied by both politicians and even economists to develop policies that can empower as many people as possible.
Competing rights can be very polarizing e.g. in the United States, the right to bare arms is contested by many, who view it as a dangerous act that might endanger the lives of others