There's a whole bunch of pics of her riding here:
http://www.hondapcx.org/viewtopic.php?f=18&t=973
She rides tunnel bag, with a breakaway clip for safety. She sits firmly between my legs, and if I have to do any advanced maneuvering to avoid obstacles, I can hold her even tighter.
It's a silly pic because she's blinking, but I think this one best shows how we typically ride:
It's certainly safe for me. I don't ride any differently with her than otherwise, I'm a very defensive rider and ride even more vigilantly with her on there. I love my pup, I wouldn't ride with her if I wasn't confident in my ability to keep her safe.
I'd say she is as safe as I would be without a helmet, which is... relatively safe. They don't make functional dog helmets... She's a smaller animal than we humans are, and therefore has greater relative bone strength, but still I'm not exactly content with the lack of gear she wears. There's nothing functional for her to wear, so it's an accepted risk for the amount of fun she has. Behind the windshield, she is 100% protected from the elements, and has no risk of getting hit with road debris. She has great fun riding -- if she gets in the garage, she goes and sits on the scooter and looks at me. She loves to ride.
As for other people on the road, I don't see how they're affected at all. Maddie is well-trained, and trusts me completely. I doubt she comprehends the danger associated with riding, but she understands that it's serious and that she has to act disciplined. For example -- she stays seated at ALL times firmly between my legs. She never stands. If she has to adjust, shake, pee, poop, or if something's wrong, she licks my arm. I pull over and stop. If she stands and looks at me when we're stopped, she has to go potty. Otherwise, she'll adjust, itch, or whatever, sit back down, and look at me like "why are we still stopped?" She has fallen asleep before on long rides, which I think is the only time she'd ever be at risk. That said though, I have a backpack she rides in at night so she can sleep:

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When we use the "maddiepack," I usually put her on my front and zip her into my jacket.
My dog has ridden more miles than most scooterists. On the trip in this thread alone, she rode just under 1200 miles. She's been riding in one form or another for several years now.
I have no idea of the legality of it. I've never been pulled over, but I've talked to cops at gas stations and at rallies she's been to, they're always very impressed and have never mentioned any legal issues associated with her riding.
Hopefully that clears it up some for you. I know some people think her riding is inhumane (I've had to defend myself once from a drunk woman at a rally), so I tried to explain as best I could for any other skeptics out there. If you want more clarification or info on us riding, feel free to ask.
