As private companies, do they fall under the jurisdiction of the United Nations Outer Space Treaty?
All of the leading private space companies are incorporated in the U.S. (e.g., SpaceX, Orbital ATK, Sierra Nevada, Blue Origin, ...). They are subject to state and national law, and also to any international treaties to which the U.S. are party. So yes, these companies fall under the jurisdiction of the Outer Space Treaty.
None of these companies want to reincorporate outside the U.S.; doing so would risk their contracts with NASA and the DoD, risk their ability to access classified and sensitive but unclassified information, and risk having their intellectual property confiscated on the basis that it is based on U.S. classified / sensitive but unclassified information.
Right now, there's no risk of this happenning; those companies need access to U.S. launch facilities. The future may be different; there's already a bit of tension in the deliberateness of how the U.S. government operates and the speed at which those private space companies would rather operate.
Is there any way the international community could prosecute them for contaminating another world?
That's new territory. However, companies that have committed crimes against humanity have been broken up and have had key personnel jailed by international tribunals. Companies that violate international treaties in a lesser way are usually dealt with in the country in which they are incorporated or have their headquarters.
The only source I could find was a statement from the senior director of communications saying "SpaceX takes planetary protection very seriously", but this yields almost no useful information.
SpaceX apparently will take planetary protection very seriously for its unmanned missions to Mars (e.g., setting up protocols, ensuring the vehicle is assembled in clean room conditions, performing additional sterilization actions between Earth and Mars). However, SpaceX is very serious about sending humans to Mars. Humans on Mars is game over with respect to planetary protection. How SpaceX and NASA will address this is a problem that is apparently being left to the future.