I've never felt dog mesh was a problem, I steer round it. And I don't think I've often felt I was about to be attacked by a dog. My worry is mainly that a loose dog (or one on a long lead), will turn across my path suddenly and we'll collide. I don't think we can expect many dogs to have any sort of 'road sense' especially on a shared path, where there's no kerb for them to recognise - hence I always slow down when passing a dog.
A while back TheDoc and I were riding back along the York-Selby path and a little fuzzy type trotted in front of us and along the path. We both slowed and moved to one side to overtake. It looked back at us, and moved over, the same way. So we moved the other way, and it did the same! It was getting quite funny, every time we moved, it did too, just enough to dissuade us from passing. Its owners were up ahead and when they realised, they apologised and grabbed hold of it, and we just laughed about it. It was just like that dance you do when trying to avoid bumping into someone coming down a corridor...