It is simple and natural to expect that private police services would tend to develop their own private judges, and vice versa. The simple way police is ensured needs no explanation : consumers would for instance buy small emitters allowing private patrols to quickly identify with masks or glasses their customers and care solely for them (except of course when there are economies of scale in indiscriminate protection e.g. terrorist attacks).
Cooperation between private polices, as well as cooperation between private judges, is a thing the market can be expected to deliver without difficulty. Mutual recognition is a simple principle. Companies have a very low competitive incentive to oppose the decisions of other companies – except when the whole public can be expected to agree, an extremely rare case in which absolute freedom to wear weapons happens to contribute to the social order, but having a company so corrupt that it requires such kind of stand-off is extremely, extremely unlikely in the spontaneous order where corruption is much more likely to be caught much upstream ; and absolute free market means users will have all time to bankrupt the company by simply leaving it.
Hence the only process that can be expected is a process of constructive competition in the framework of the very obvious cooperation agreements (i.e mutual recognition of the decisions that will, obviously, be a key criterion of decision for consumers).
Hence there is no need to fear a situation in which the absence of a centralized body for justice would mean constant discussion of decisions. It is also highly likely that judges will want to share as much files as possible publicly. Why ? Because of course judge is a career where your honesty is the main reason why you can be expected to be picked by new customers – hence transparency is interesting for them.
Mutual recognition is simple as the basis for the legal code is as well simple : absolute respect of the human body (dealt with as the property of the person) and of all the other private properties of the individuals, respect of private contracts. Common sense (for instance understanding that if proof can be brought that an individual has been manipulated, his decision cannot be really seen as his own). Understanding that there are some human activities that no one can be expected to be willing to undertake fully of his/her own will (I see a single one : selling his/her own body to dozens of customers everyday for sexual intercourse – to quote Mises in Socialism “prostitutes belong to the owner of the brothel”).
As explained in an earlier article judges could pay themselves by levying money on the culprits and free market will fix their income but judges could also be expected to become workers for security companies encompassing police and justice (as pointed out earlier). In this case the company seizing the suspect (free competition will be the best incentive for absolute respect of the human rights of said suspect) will have the hand in dealing with the enquiry, agreements between companies lead to recognition of justice decisions, and the judge will be paid through the subscriptions of the customers. This is certainly a much more efficient way of dealing with security.
As always the fast information network of nowadays (companies like Twitter for instance) allow for the quick discovery of issues (scandals…) and thus re-adjustment of market demand according to the possible failures of service providers.
For defense, as already pinned out it is clear to me that nuclear power makes it very cheap. It is a charity work some big company would want to ensure (to any area it wishes – no need for borders of any kind, it is obviously one of the actual purpose of abolishing the State) for self advertisement, while reducing spending by adopting all-out retaliation instead of gradualism to better discourage offense (for instance “Apple decides to shield California and Oregon under its 5 nuclear armed stato reactor powered cruise missiles” while “Dassault and EADS cooperate in offering to Western Europe nuclear defense with two nuclear submarine drones armed with SLBMs”, etc – it is a purely voluntary process, and economic calculation would drastically increase efficiency and lower costs – how do we need thousands of H-Bombs when a very, very limited number are enough to threaten with decapitation)… Also the disappearance of the State ought obviously to lead to the actual climate of peace needed to limit costs even more. In general : how come a single company be expected to want to start any war ? How could it fund something so deadly and costly ? And permanently alienating a big slice of the world’s consumers ? Even if (as can very legitimately be expected) the areas without State are a limited part of Earth. There is no single person that has any interest in putting his own life at threat in any war he/she would certainly be found responsible of, and attacked for. Doing this (building ICBMs, for instance) while hiding is, impossible, simply put. Business has all interest in protecting its own assets and customers against the invasion of States ; so business does the work.
To conclude I will just point to the Lipsey Lancaster paradox, as I can be expected to do very frequently now : it is the mathematical proof that so long as you don’t eliminate 100% of the Leviathan (the State) that disrupts the ever-spinning economy (Mises), it will regrow and keep sucking the energy out of your business (through intimidation with regulations and standards, or through the oligopoly of banking creating inflation, or through taxation, or maybe even through the military draft) – the State exists solely through the exploitation of resources produced by others. The $21 trillion hole in Pentagon accounting hidden by the US Treasury (Michigan State Univ research) is here to testify of it. Depleted uranium is the proof they are experts in lying of the same level than the Pravda (visit http://www.depleteduranium.org). They will never change. Politics will be always for vampires. Eliminate all means they have to interfere, I mean all (remember again the Lipsey Lancaster paradox), or they will always find a way to turn your life into an inferno.