I think it will be very similar to how Antarctica is today. There is a good starting point as a study case.
Several science bases (maybe even some civilians, as Chile have in Antarctica today), each base belonging to a country of origin on earth, so each base will have a "local" law depending on that country. Lots of interactions/helping between bases.
Mars won't be considered a foreign territory by any country, for many years to come. As in Antarctica, claimed territories will overlapped, but should not be a problem for anyone for some time.
Problems will start when countries start exporting natural resources back to earth... specially coming from those overlapped territories.
Thats my 2 pesos.