What happens on mars stays on mars?
I suspect that if and when people eventually get to Mars and get to stay there, the various stakeholders (earth governments, blocs, and corporations) will initially cook-up something similar to the Antarctic Treaty  that has regulated human conduct in Antarctica for the last sixty years. And that'll stay in-place until it hits it's scaling limits, whereupon it'll be replaced by something with more involvement from local people and institutions. What that would look like, I don't know.
Geography is history and Mars isn't anything like Earth, so it'll have its own historical development. The people there will still be people, but they'll be changed by their experience. I'm not sure that templates from Earth (perhaps even including the concept of constitutions) will be useful beyond the short term.
However, given that no space-faring nation (U.S., Russia, China) has ever ratified the Moon Treaty, and, given that the U.N. normally can't even enforce its directives here on Earth (especially not if a Security Council member -- such as, again, U.S., Russia, China -- objects), it would appear to be rather toothless.
In some place like this
they throw you outside if you can't pay your air bill. We've shown people who contaminate the atmosphere on Earth a lot of forbearance (e.g. it takes 100 years of CO2 emissions to create a crisis), but if you contaminate the internal atmosphere of a space habitat you can endanger other people immediately and it won't be tolerated.
With the right kind of technology base it might be different: say you had some kind of self-assembling machines that could build small habitats that are autonomous. You might find small groups of people who go there own way and assemble with likeminded others.
>you could possibly be sent back to earth?
I'd guess that'd be a bigger cost than that of keeping Billy (the prisoner to-be) in the habitat, with damages and loss of working man-hours included.
I presume the colonists agree on what'd happen if anyone became a problem in the colony before they set off, but just for thought's sake, if we go into the costs-first way of doing things then it wouldn't be out of the question that the other colonists would just kill Billy, or put him into some other not-alive state:
- Putting Billy into some sort of prison would be a loss of food, man-hours (he's no longer working and someone must keep tabs on him, feed him, and stuff) and space, all precious commodities in Mars.
- Exile would mean certain death without giving Billy a surface-suit (big cost, not replaceable), and even with one it'd also mean an even worse fate in the barren landscapes of Mars. Knowing there's nothing out there for him just means he'd stay around banging at the windows or being an active threat to the survival of the colony.
- I mentioned shipping Billy back in the earlier paragraph.
- Using Billy for compost would be extremely unethical and life-scarring for the other colonists but also the only (I think) cost-positive way to put an end to an irreversibly disruptive colonist. Key-word is 'irreversibly'.
- Rehabilitation is a problem we haven't solved here on Earth yet, hence it's not mentioned as an option, but if you think it can be done please say so in the comments.
The morals of this are... don't optimize for costs, and talk things out with those you have a problem with.
A Starship that goes to and from LEO orbit might be reused once a day or more. In the time it takes to make the transit to Mars a "local traffic" Starship might be reused hundreds of times.
Thus in terms of capital intensity there isn't much difference between using and reusing a martian Starship. In fact it might be better to break it up at the destination, live in the tanks, recycle the metal, etc.
We are sure now that there is water, carbon dioxide and possibly some kind of ice in craters near the lunar poles. In fact I am pretty sure a chunk of water ice could be stable 500 m under the moon’s surface.
It probably adds up to a medium or large lake which would be a good start for a closed ecology but could be used up quickly if people used it to make fuel.
Ceres has large inventories of volatiles on some sense. You could support a population of 1 trillion+ people in something like the O’Neill colonies.
In fact ‘small ringworld’ structures could be built with possible materials, especially if some kind of journal bearing could be built that handles 10+ km/sec velocity differences to transfer centrifugal stress to a reinforcing band. (Magnetic?)
That kind of structure wouldn’t need a roof; landing on a rotating ring about the radius of the earth (but inside out) would be about the same as landing on the Earth in terms of atmospheric entry.
It leaks gas though. A space-based civilization could last for millions of years if the rate of volatile leakage could be kept very low. Something like ceres looks like a huge amount of material for a human lifetime but a civilization built to last might take extreme measures to conserve volatiles. (E.g. reaction fuels might be unthinkable for routine use.)
At first, everyone on mars is going to have family and assets on Earth, and thus earth based governments are going to have a lot of leverage. Of course oversight will be difficult, so if a company wants to do something illegal it could probably get away with it, but they'ed be more like mafiosos than royalty. I doubt we'd see any abuses on the same scale as say the east india company. After a few generations though there might be a sufficiently large population with sufficiently little connection to Earth that they may want to come up with their own law system. It's unlikely that any mars colony will provide much value to earth beyond national pride, so by that point they might face little resistance so long as they remain nominally under their parent nation's jurisdiction.
Realistically, corporations will probably get to Mars first, but governments won't be far behind. I would recommend keeping your rage-killings to a minimum for that reason.
I don't think you have to go to Mars to do this. Many factions have claimed convenient islands on Earth as their own and have built highly functional societies even though other entities claim that they're the actual government. But of course, those societies watch the power struggle very nervously. If you can do it 80 miles off the coast of China, you can probably do it 230 million miles away.
Both constitutions and monarchies are choices. They can (and should be) legitimized by a souvereign.
You could argue that once people live on mars, they can choose to establish some sort of council that designs a constitution.
By the way, constitutions are typically informed by existing law, so it is plausible to assume that any mars constitution would resemble the (modern) ones we have.
 To add: Of course anybody can claim some constitutional basis to exist, but everyone can challenge that. See Russia's occupation of the Krim peninsula, which quickly turned into a legal battle regarding souvereignity.
Is there a need for a constitution? Early Mars will be dominated by high productivity. You need someone to build and do the things. Recreating massive numbers of vertical industries is impossible to do without people being in top productivity. There wont even need to be a union. The 'king' will be required to keep those people happy.
If there's even a slight threat of being thrown out an airlock, people wont move there. You will fail early. There wont be a need for a constitution for quite some time.
I suspect it'll become a chinese land, since they'll be there first
We are too busy trying to manipulate and profit from each other to even start to think about that kind of detail
One of its provisions says, "If you happen to grow a third nipple, expect to have your air supply cut off."