Here in Oklahoma, you can't take your dogs to the state parks and/or equestrian trails unless they are on a 10' leash. That makes trail riding with a dog in tow difficult at best. We did take our old "Blue" with us when we rode around the 2-mile section. She would poop out on us after about the first half mile or so and head back for the house. We didn't have any dog issues, with or without her.
The dogs we have now I would not take anywhere on horseback. The daughter's two big dogs are undisciplined and uncontrollable.
She keeps them chained up all the time or they would run the horses to death. Our two dogs are kept in a fenced in yard when they are not in the house. We let them out to meet the horses when we first offloaded them. Husband's big boxer/pit mix panicked and ran back onto the porch and hid under the porch swing.
My little Llasa/terrier mix sniffed a few times and got bored with the whole thing (he's old). He does love going poo hunting, though! (Big dog has since gotten used to them being around and will sniff noses with them, but she still runs to the porch if they so much as sneeze in her direction, the big coward!)
As for bears - I would rather imagine the big mean looking boxer/pit mix would head to the next county if we ever encountered one. The little Llasa/terrier mix thinks he's a rottweiler in disguise and would probably try to tackle the bear. I think we will just leave them at home...