It would be impossible for me to do this experiment. I toss and turn so much when trying to get to sleep and fall asleep so gradually that by the time I settle into one position my brain’s already too asleep to register what it is.
Likewise, in the morning I can’t initially think straight enough to notice what position I’m lying in (by the time I can, I’ve already rolled over or gotten out of bed completely). In all my life, I’ve never once noticed what side I woke up on.