Funny you should mention your dog, and something crouching on your chest, Rev Blair.
I used to have these experiences quite often too a few years ago. Always after around 4am, always when I was asleep on my back. They were just as terrifying as everyone else has reported. As they became more familiar I found that I could force myself to scream, or at least so I thought. My g/f told me all she heard was a strange whimper.
And then one day the presence (there was always a feeling of a presence and it seemed different each time) turned into a dog, and instead of the usual malevolent feeling I suppose something in my brain associated dogs with good feelings, and so the dog jumped lightly onto my bed and licked my face. I woke up laughing.
Ever since then the experiences have been few and far between, and never as scary as they were, and I always had enough willpower to force myself onto my side so that I'd wake up.
Incidentally I also figured out how to induce the experience. If I lie in bed after waking up, for about 30mins or so, and I don't allow myself to go back to sleep for that time, I start to feel a little groggy and drained. If I let myself go back to sleep then I'm almost guaranteed to wake up in sleep paralysis soon after.