I use to with my old mare, but I stoped doing that when a woman brought her dog out for a trail ride and her dog ended up attacking another persons horse, if you can not control what's going on with your horse and your dog it's a bad idea!!
Oh my goodness that is really funny about the saddle bags! I was going to take my pup in a backpack when she was a puppy, of course she was like 6 pounds and slept all the time! I didn't do it though!
I agree about keeping control of the dogs. I take her on trails we are not likely to run into anyone else. I keep her on a leash in the parking lot and walk both my horse and dog out of the lot. I get off fire and walk both of them if there are others we run into on the trail. I hate it when dogs go runing all over and don't care about what is going on!