New here, but wanted to comment on this topic. Locally, we have lost sections of privately owned trails due to dogs. Apparently a dog ran into a yard and started a fight with a landowners dog, so the land owner decided to no longer lease the land to the trail system.
While I have seen very well behaved dogs on rides, my honest opinion is this is horse time and dogs don't need to be there. They can go on their own walks at appropriate places or dog parks - it's a commitment we make when we have dogs, so we make time for all our animals to have their time.
There are a lot of horses who dislike dogs and will hurt/kill them. Mine wasn't one of those until a JRT jumped up and bit her nose when she was trying to be friendly. She's better now, but there for a long while, I was afraid she would seriously hurt one.
We as riders really need to start thinking outside of "ourselves" and realize when we are in public areas, it's not just about "me and my horse" or me and my group. Same with littering, drinking and noise level on trails. I can remember a huge group of riders who even packed in a "boom box" with all their beer and food and aside from blaring music when I really wanted to listen to the river I was riding beside left every bit of trash on the trail in a state park.
There's also the safety issue for our dogs. I just don't believe in bomb proof horses (even though most would say mine is) or 100% trustworthy/trained dogs. I had a dog that stayed at my side like glue and wouldn't go in a road without being on leash - she was killed by a car. It just isn't worth it to me. I get crap from a trainer I know about always having my dogs leashed when they're not in the fenced yard, but I'd rather a live leashed dog then dead free one, and you can't predict a dog's reaction to all stimuli. Well especially since I have Beagles - they aren't hunters, but that instinct is there.