They are indeed called "door check straps" your doors should have them. Here is some pictures online where I have circled where they should be attached and what one looks like. These pictures are supposedly taken of a 2005 JGC door. As to why your passenger door doesnt have one, Im not sure. Could be broken too, but less noticeable because the passenger door isn't used as often as the driver and the hinges haven't worn out from the excessive stress. Feel along the door, is there an opening to no where? We have cars come in with broken check straps that retract back into the door itself.
"so i did a full tune up soup to nutts."