Does running Google Photos in addition to the iOS Photos app cause two copies of each of my photos to be stored on my phone?
The photos are stored on your device and Google accesses them there. 

It makes a copy of the selected photos (or all photos, depending on your settings) and uploads that copy to their servers for storage. 

You only have 1 copy of the actual photo and not your device but some space is also used by the Google app to keep the links between your photo and it's databases in sync.