Picture if you will this scene:
It is about 4 am. An exhausted mom slumbers in her bed. Surrounding her are the family’s two dogs, cuddled as close as can be on either side, and also sleeping peacefully.
Enter one very frightened six year old who bursts into the room!
I’m not sure if she scared the dogs, or if the dogs scared her, but Belle was screaming and the dogs were barking and the sleeping mom (that was me for a time) was awoken with a start.
Belle had had a bad dream and wanted to sleep with me. I sleepily agreed, and we all, people and dogs, settled down to go back to sleep. We all got up normally (me a tad less energetically than usual). Belle was her usual on-fire self and there seemed to be no lingering affects from her dream.
Until her sister got up.
The first thing Belle said to Krystal was “I slept with Mom last night.”
Krystal called her on her motives immediately, “Are you saying that to try to make me jealous? Because it’s not going to work!”
Why oh why is everything a competition with these two?
Niceties and Nasties from New England 
I suppose you may have thought of this, but I’ll offer it up as a suggestion, anyway. When one or the other complains – is competitive – try to glorify the position of the lesser. That is, if one slept with you, and the other see it as a treat, outline the great advantages of having the rest of the house to herself, etc. Just a thought that may be helpful.
As for the root of the problem, the competitiveness itself, I can offer no help. Being an only child, I never had to deal with it.