Getting hit on the right side of the car is often called a passenger's broadside.  This can cause the head to bend suddenly to the right.  In someone looking straight ahead, this bending can lead to over stretching of the left upper trapezius and levator scapula.  Also, the facet joints on the right can get jammed suddenly, leading to a right sided facet syndrome.  More rarely, the head can bend far enough to cause a sudden closing of the intervertebral formen on the right and bruising in the spinal nerve roots with numbness and tingling down the right hand.  

It's also common to see the passenger sustain direct trauma to the right shoulder, rib cage, and AC joint.  The head can also hit the passgener's side window, leading to a concussion.  Right sided elbow injuries are not uncommon.

If the passenger has his or head turned, this can cause either more (turned to the right) or less (turned to the left) of a facet component.   Having your head turned right can also lead to an injured left SCM.   When the body is turned right, the sudden right side bending of the rib cage and thoracic spine can lead to bruising of the right rib cage. 

Finally, the right hip and pelvis of the passenger can be shoved into the door (or extensive door crush can provide a forceful blow to this area).  This can lead to an over stretching of the right SI joint ligaments. 

The driver can sustain all of the forces above, but usually doesn't hit the passenger's door.  Stick shift or console injuries on the right side of the driver's body are not uncommon.  The right knee can sustain a medial meniscus (cartilage) and MCL (knee ligament) injury as it hits the gear shift or console (similar to getting tackled in football).