The Human Rights Act allows people to bring a case only against an organisation that is a public authority. So a person who is employed by, for example, a local council can take proceedings against their employee, but a person who is employed by a private company cannot.
Even so, the Act affects court cases between individuals and private organisations. This is because it changes the way the courts interpret and develop the existing law. The courts are already using Article 8 of the Convention (the right to respect for private and family life) to develop a law of privacy that will affect private individuals and organizations as well as public authorities.