I am a cat person myself. Though I've had and loved both dobermans and pugs. I never let them out un-leashed.
I feel I should say this, though it might not be popular amongst dog lovers- if I were riding my bike and I saw a dog (especially a rottweiller), off-leash, even near its owner, it would be very unpleasant for me. Many dogs hate bikers, and I would be depending completely on a stranger to control their off-leash dog. I would have NO way of knowing whether the dog was friendly or unpredictable, or whether the owner was responsible. That's a creepy feeling.
There is a boxer chained in the front yard of a house I pass regularly on my FAVORITE beautiful bike ride. He barks, snarls, and leaps into mid air against his chain and leather collar when we pass. Yes, he is chained up- but I live in fear that his collar or chain will snap one day while I'm passing. Many people have had bad experiences with dog attacks, and are justifiably fearful of loose dogs.
I think when we are in public places we should take other people's feelings into account as well as just our own pleasure. People like to feel safe on public roads and trails, in public parks, etc.