The biggest issue with such systems is that they do not provide the stability we typically seek from legal systems. Accordingly I would expect the set of laws accepted by all such groups to handle the vast majority of day to day cases.
The purpose of laws is to provide either a codified document of what is right and what is wrong, or a codified document of causes and effects, depending on how cynical you are. It is designed so that the rules are known ahead of time. You know that if someone steals your wallet on the street, there is a system of justice which will seek out the thief and run them through a particular set of rituals, and possibly do something like incarcerate them.
Now it's obviously more complicated. We don't know all the laws and their nuanced details. That's why we have lawyers to sort things out when we finally get to the court proceedings. Your system would need to have a remarkable caste of lawyers to untangle it, especially resolving any inconsistencies which may arise.
However, it is not the lawyers that I want to focus on. You and I have an intuitive understanding of our particular legal system. We understand that if we do certain acts, that will be deemed "murder" and the justice system will be swift and servere. If we do other acts, they will be deemed "speeding," and the justice system is a bit more murky regarding that. There are plenty who will make informal arguments along the lines of "5mph over the limit isn't really speeding" or "I was just going with the flow of traffic." But in all cases, we develop an intuitive sense.
This intuitive sense is essential for making rapid decisions. The faster a situation changes, the more intuitive one needs to be. If I am an armed citizen of group X, and I have a law which says "I am allowed to shoot people who break the laws they abide by," and I see someone of group Y draw a weapon, I may have a split second to decide whether I should draw my own weapon, or if I need to back down because that particular action was legal by Y's rules.
The pressure to make these intuitive judgement calls will rapidly encourage the governments to generally unify the laws which may have effects on people making rapid decisions. Generally speaking, you'll find the rules for whether or not a person can have a firearm will become clearcut such that all citizens can intuitively respond to someone drawing a firearm in a legal and appropriate manner. It may be funny ("only people with red hats are allowed to have guns," and a strict ban on wearing red hats by anyone else), but the resulting agreed upon rules will be something that the members of all governments can intuitively understand and operate under.
This pattern will not necessarily happen for aspects of life which are slow enough to think through. In those cases, each government may indeed have their own laws. I came across a fascinating example recently on the Christianity Stack Exchange, where a Catholic was asking whether his Buddhist fiance could be baptized while keeping her faith. The answer was a resounding no, but further digging showed that the real issue was that he wanted the marriage to be valid in the eyes of his faith, so that he was not living in mortal sin his entire life.
To that new question, people were able to provide positive replies. The "lawyer" caste of the Catholics indeed had a process to permit a Catholic to marry out of their faith. They simply had to apply for a "dispensation for disparity of cult" from their Bishop. The Bishop could then choose to grant it, and generally speaking such dispensations are granted because it's an opportunity for the Catholic spouse to help their non-Catholic spouse choose to embrace Catholicism under their own freewill (rather than under the duress of "convert or we can't marry").
In the case of marriage, this is not a "spur of the moment" decision. Accordingly, specific nuanced laws like this dispensation for disparity of cult (that I had never heard of until I read the question) will probably survive in such a multi-government environment. Indeed, one could consider this specific case to be a practical real life example of such a multi-government system operating.
Sometimes this will affect the social dynamics. Consider a case with corporations. My employer has told me some particular piece of information is proprietary. You approach me, asking for it, saying "the laws of my employer permit me to know your employer's proprietary information." This can happen in real life situations. As such, it is now socially acceptable for me to slow this transaction down and get time to think. I'll say something along the lines of "Let's have your boss talk to my boss, and if our HR departments and general consuls agree that it is acceptable, I'll tell you that bit of information." Both of us work into our intuitive understanding of the system that it is acceptable to slow the process down to "thinking speed" and make sure we were legal. (And a great deal of the art of social engineering is how to prevent people from doing this)
