Yes, as citizens of their country they have to abide by international law. The law doesn't just apply to government based missions but to all citizens of that country.
I think the main reason why they might not get permission to go to Mars is because they would have to show that their colony won't interfere with scientific explorations of Mars.
There are many potential habitats on Mars. None confirmed yet but that's because we haven't looked in the right places yet with the right equipment.
If these habitats exist they won't be easy to confirm from orbit, and their habitability very hard to assess from orbit.
And the problem with a human mission is that, sadly, humans can't be sterilized (plants can be, by sterilizing the seeds). If only we were plants... But as animals, then we carry hundreds of trillions of microbes wherever we go in tens of thousands of species. And though many of those microbes would have no chance of surviving on Mars, they also include extremophiles, microbes that can survive almost anywhere - because you find those everywhere - even in soil, clothes, food, in and on our bodies, and covering surfaces of human (and robotic) spacecraft. We can sterilize robotic spacecraft but not human occupied ones.
So before a mission like that could be approved
a COSPAR workshop would need to show that it is consistent with planetary protection requirements, and would not risk introducing Earth life to Mars surface habitats.
Or else there would need to be international agreement that Mars no longer needs to be protected from Earth microbes.
Otherwise, of course, we risk going to Mars just to discover life that we brought there ourselves. What could otherwise be the greatest discovery in biology could turn out to be a huge anticlimax.
So, to my mind, seems unlikely that either could happen before the 2020s or 2030s. Because Mars is so complex, so many places to explore for possible habitats, because of the global sandstorms that can carry resistant spores throughout Mars, and because it does have some potential for habitats. And if there are habitats there, the biological effects of a human landing, especially crash landing - would surely be impossible to reverse.
But we could send humans to explore Mars from orbit. And they could plant flags on its two moons, Phobos and Deimos probably without any planetary protection issues. Also of course send more flags on robotic missions. See To Explore Mars with Likes of Occulus Rift and Virtuix Omni.
I think they would just not attempt to go, if the mission was shown to be likely to interfere with scientific exploration of Mars, and COSPAR didn't approve it. But if anyone did try, they would need to get permission from the launching nation and also their government would keep tabs on what they are doing, and the people who supply the rockets and other essential equipment would also have responsibility too.
E.g. if the US supplies rockets (say through Elon Musk) to launch a spacecraft for an organization based in another nation (e.g. Netherlands) and the spacecraft is launched say from Argentina (List of rocket launch sites), and has nationals in it from Japan, China, India, or whatever - well that just adds to the number of nations responsible. All those nation states would be involved in the project and responsible for working together to uphold the Outer Space Treaty.
In the same way governments are required to keep tabs on movement of potentially invasive exotic species around the world and to prevent their citizens from evading quarantine regulations and to collaborate with each other in enforcing these requirements.