Section 87 of John Locke's 2d Treatise on Government: “Of Civil Government” (in Chapter VII, "Of Political or Civil Society") summarizes John Locke's contract theory much better than many summaries I have seen by others:

Man being born, as has been proved, with a title to perfect freedom, and an uncontrouled enjoyment of all the rights and privileges of the law of nature, equally with any other man, or number of men in the world, hath by nature a power, not only to preserve his property, that is, his life, liberty and estate against the injuries and attempts of other men: but to judge of, and punish the breaches of that law in others, as he is persuaded the offence deserves, even with death itself, in crimes where the heinousness of the fact, in his opinion, requires it. But because no political society can be, nor subsist, without having in itself the power to preserve the property, and in order thereunto, punish the offences of all those of that society: there, and there only is political society, where every one of the members hath quitted this natural power, resigned it up into the hand of the community in all cases that exclude him not from appealing for protection to the law established by it. And thus all private judgment of every particular member being excluded, the community comes to be umpire, by settled standing rules, indifferent, and the same to all parties; and by men having authority from the community, for the execution of those rules, decides all the differences that may happen between any members of that society concerning any matter of right; and punishes those offences which any member hath committed against the society, with such penalties as the law has established: whereby it is easy to discern, who are, and who are not, in political society together. Those who are united into one body, and have a common established law and judicature to appeal to, with authority to decide controversies between them, and punish offenders, are in civil society one with another: but those who have no such common people, I mean on earth, are still in the state of nature, each being, where there is no other, judge for himself, and executioner; which is, as I have before shewed it, the perfect state of nature.

One of the remarkable assumptions that John Locke makes here, and in the 2d Treatise more generally, is that a government cannot have any power that individuals didn't have to begin with. This is a line of thinking that my fellow University of Colorado Boulder professor Michael Huemer pursues in a fascinating way in his book The Problem of Political Authority: 

To the extent I think that governments can legitimately do more than Michael Huemer thinks, it is in part because I think individuals in the state of nature also have the right to do more. For example, I consider individuals in the state of nature to have some rights to compel others to contribute to public goods, especially in exigent circumstances. 

What John Locke clearly states is that governments can put people on trial and punish them if found guilty because individuals had the right to put people on trial and punish them if found guilty in the state of nature. The great virtue of having governments put people on trial is that it is very difficult for individuals to see clearly when someone has done them wrong. I talked about this in "John Locke: People Must Not Be Judges in Their Own Cases." In application to present day circumstances I emphasized in that post the problem of cases with an administrative agency on one side and an individual of company on the other that are judged by judges hired by the administrative agency. A similar problem arises when a company is able to get away with mandatory arbitration in which cases are judged by judges hired by the company. This violates the sound practical principle of justice that people must not be judges in their own cases just as much as administrative agency cases being judged by judges hired by the administrative agency. Thus, in my view, the Consumer Financial Protection Bureau was upholding a Lockean principle when it ruled against mandatory arbitration, while Congress and President Donald Trump were anti-Lockean to overturn the rule against mandatory arbitration.  (I use principles from John Stuart Mill's On Liberty, John Locke's 2d Treatise and Robert Nozick’s Anarchy, State and Utopia to analyze the legitimacy of other aspects of the Consumer Financial Protection Bureau's role in "On the Consumer Financial Protection Bureau.")

The social design problem of organizing things so that our biases don't cause us to do things that are unfair comes up in many other contexts as well, that are less political. For example, economists who are members of the American Economics Association who care about justice have an interest in refereeing systems for the American Economic Review that minimize bias. One thing I worry about in that context, for example, is that the fear that an editor of a major journal might take revenge on someone may inhibit referees from being as tough on the paper of such an editor as on the paper of someone who is less powerful. This is not exactly being a judge in one's own case, but it presents a similar problem of bias. Here of course, I am taking as given the fact that it is very hard to hide the identity of a referee. Even if one avoids citing one's own work in a referee report, the society of economists is in many ways a small world, so that one's views and writing idiosyncrasies are often well known.  

To come full circle, note that one right people have in the state of nature that we don't give up in our polity is the right to criticize others. Here we often organize socially to do collective criticism, as in an election or a poll (where private enterprise is doing some of the social organizing), not because we are giving up the right to criticize as an individual, but because we are afraid of reprisals if we criticize powerful people as an individual, without some protection of anonymity.    

For links to other John Locke posts, see these John Locke aggregator posts: 

• John Locke's State of Nature and State of War
• On the Achilles Heel of John Locke's Second Treatise: Slavery and Land Ownership

Also, see these other posts referencing Michael Huemer:

• Michael Huemer's Libertarianism
• Michael Huemer on Moral Progress
• Michael Huemer's Immigration Parable
• "The Hunger Games" Is Hardly Our Future--It's Already Here
• Does the Fabric of Our Society Depend on a Lie?
• An Experiment with Equality of Outcome: The Case of Jamestown