I've been having to deal with this sensation since my first panic attack a year ago. I would also like to note that I have had a few concussions in the past, one being pretty severe and affecting my cognition for a little over a year.
When I sleep on my left side I feel fine and can fall asleep relatively quickly. However, when I sleep on my right side I get a very bad headache as I sleep. It feels like a pulsing in the left, front top side of my head, just above my temple. It feels like blood flow is constricted. If I want to lay on my right side I need something to be touching the left side of my head, and/or I'll pull my hair on the left side which often relieves the intrusive sensation.
Could this be something to do with nerves or cranial fluid pressure. My doctor always brushes it off when I tell him about it, it may simply be health anxiety.