My steel frame Ironman is flexy on the Cycleops trainer. No big deal, it's fine on the road.

I suspect a more secure trainer would reduce the side to side shifting. The Cycleops and similar trainers use a fairly primitive threaded adjustable bolt to support one quick release end, and a sliding locking clamp on the other side. The cups that hold the ends of the quick releases allow a lot of slop. So it's not so much the frame flexing as the entire bike shifting on the trainer. And there is some frame flex, especially the bottom bracket. Trainers with better supports probably shift less.