When hanging a double layer of drywall, do both layers need to be fully screwed to the resilient channel?
Yes, both sets of screws are required. 

Two layers of gypsum board is generally used for two hour fire wall protection. The protection (and rating) comes from the proper installation, including proper nailing for both layers. If a fire develops and causes the top layer to fall, then the second layer would not have approved fastening to develop the two hour rating. 

If the second layer is used for sound control, then no, the additional fastening is not required. (Sound control is developed in the material, not the fasteners.)