ARS §28-735 describes the law for motor vehicle passing of bicycles, ARS §28-812 which describes the cyclists legal responsibilities on the roadway and ARS §28-815 describes where bicycles can operate on roadways.
The bottom line is that cyclists must stay as far right as is practical and safe given the conditions of the shoulder or bike lane. The cyclist can ride in the roadway if the shoulder or bike lane is in an unsafe condition, which they frequently are. The motorist has the obligation to provide a minimum of 3 feet of clearance when passing the cyclist. The cyclist is under no legal obligation to pull over for you, it is your responsibility to avoid him.
You are not surrendering your rights by sharing the road with cyclists, you are breaking the law by not.