It should swing. Regarding height, it can go from to shoulder height to above the head. I prefer shoulder height, because it increases the frequency of the swing and thus makes it a harder workout. If you swing to above the head, you also get a little pause in the top as the bell decelerates, pauses and accelerates back again.
The movement is driven by the hips, not the shoulders. WARNING! THE NEXT DESCRIPTION IS GRAPHIC AND ADULT IN IT'S NATURE!
Think 'naughty' when you do the hip movements. Imagine you have the longest rod ever seen, and that you need to satisfy a woman standing a meter away. You need to 'penetrate' something imaginary with your loins. The thrust you generate with this movement, will press you arm forward, your body upwards and hurl that big ball of steel up, up and away.