I believe there are different scenarios that you are describing.
First, a two lane street, traffic moving in opposite directions....one driver comes upon a vehicle or obstruction on their side of the road. To pass the obstruction, this driver must cross the center line in to oncoming traffic. The driver must wait until the lane is clear before proceeding. Oncoming traffic has the right of way.
Second, a one-way street with two lanes. Pretty much the same as the above except when passing the obstruction the driver must ensure that to pass, he or she has a safe distance in which to switch lanes. If another car is within that safe distance, the original driver must slow or stop and allow this vehicle to continue on before making the passing maneuver.
Sources: Any Driver's manual from any state or country that produces one. Do a web search.