Keep your dogs in sight and under voice control, and make sure they are wearing tags or some other identification. Most Dobes, if they don't already love snow, can learn to enjoy winter hiking. Usually if it isn't too cold for you it won't be too cold for your pooch, but if you have to wear an expedition parka it is probably too cold. The Forest Service has come very close to banning dogs on the Millcreek Canyon ski trail due to lack of compliance with the poop-scooping regulations. Bags are supplied at the trail head.
Avalanches are common in the mountains.
Dog Parks Salt Lake City has several parks which have an area set aside where dogs, accompanied by their owners, can be off-leash. Dog Parks are still in the trial stage in Salt Lake City. Whether they succeed depends on whether dog owners take responsibility and follow the rules. Needless to say, we all should clean up after our dogs. However, one of the biggest problems the city has is with people who don't observe the boundaries of the dog parks, and let their dogs run loose in other areas of the park.