What are some solid points about the topic "When the law does not succeed in punishing criminals, citizens should do so"?
As you write your essay, the first thing you need to consider is whether you agree or disagree with the statement.
In one sense, it articulates the problem that official legal sanctions are not enough to enforce behaviors citizens do not internalize. Take, for example, littering. Have you ever seen a schoolyard or classroom which smells and looks disgusting because people drop food wrappers and other garbage on the floor? How do you feel when you encounter someone's used chewing gum under your desk? Obviously, it wouldn't be feasible to arrest every kid who drops a bubblegum wrapper. Peer pressure though, by naming and shaming the slobs in your class, could change their behavior, with punishments such as community service meted out to the worst offenders. This is a positive example of how members of a community can contribute to creating a rule of law.
The other extreme is what is sometimes called "vigilante" justice where people take the law into their own hands. Perhaps the most morally heinous examples in the United States were the post-Civil War lynchings of black people in the south, often for alleged crimes that they had not actually committed. The problem with such vigilantism is that it has none of the safeguards of due process, and can harm innocent people who are simply victims of prejudice.
The notion of punishing criminals is designed to achieve three outcomes. Firstly, to provide a negative consequence for one's action. Secondly, to deter others from also committing that same crime and lastly to provide protection to the community from the actions of the criminal.
From this comes the question of whether citizens can achieve these goals or will they simply seek revenge for the crime committed.
Of course, the determination of guilt is after consideration of all the facts (legislation, witness statements, expert opinion etc). This process needs to occur in an atmosphere of "innocence till proven guilty". The nature of human emotions and dynamics would make this very difficult for citizens to decide.