My road bike (specialized allez) is a low end performance oriented bike. Based on commentary I've read here in the past I expected it to handle really poorly with loaded panniers on the back but it handles fine. It's just feels like I'm dragging somebody behind me. ;-)

As far as messenger bag movement while riding goes, in my experience it depends a lot on the bag, how you have it adjusted, and how it's loaded. If there's no cross strap, especially if you leave the shoulder strap looser, you will have an unhappy experience on a road bike. A properly loaded, high quality messenger bag with a cross strap isn't going to move much. My messenger bag was middle of the road in terms of quality and I'd get a small amount of shifting under certain circumstances. A lot of the time I didn't notice it was there but the bulkier the load and the longer the distance, the less that was true.

From a performance and handling standpoint, wearing a messenger bag had a lower impact overall than carrying the same load on the same bike in panniers.