Usually, traffic laws that apply on public roadways have no bearing on private property or roads, unless otherwise posted. While certain driving laws vary from state to state, there are specific laws about driving on private property that are applicable in all places.
Law officers cannot hand out speeding tickets on private property unless a speed limit is posted on the private property itself. In a parking lot, where the property owner has no speed limit posted, drivers can legally drive as fast as they want; however, if they break other laws such as running a posted stop sign, hitting or endangering pedestrians or driving recklessly, they can be ticketed, arrested and prosecuted. On private ranches and country roads on private property, there is no speed limit and people driving on them can drive as fast as they want at their own risk.
Law officers can only enforce "no trespassing" onto private property if there is a sign stating as such posted at all entrances to the property. Technically, people must ask permission to drive their cars onto any private property, but this usually only applies to private residences and when a sign states it at the entrance to the property itself. Those who do not ask permission before driving onto private property with a "no trespassing" sign can be prosecuted.
Laws against driving while intoxicated are not limited to public roads, but to all roads, private or public. While people driving drunk on their own private ranch land can probably get away with it since there is little chance a police officer will be monitoring their driving habits on their own private property, police officers are allowed to sit outside of bars and watch people drive out of private parking lots and arrest them for driving under the influence.
On ranches and other rural private property, minors without a state-required driver's license are allowed to drive vehicles and dirt bikes on the property. There is nothing law enforcement officials can do about this unless a serious injury or death occurs as a result, in which case the state could charge the property owners, guardians or the parents with negligence. However, unless the private property owner asks law enforcement officials to help keep minors from operating vehicles on his land, this is legal.