The dogs don't care. They want their walk.
Here's my usual dogwalking outfit:
When it gets to 45 below, however, I add a facemask and double the gloves on my hands.
Sometimes I try to be tougher than I really am, and opt not for the attention-grabbing frostbite-saving cover of a facemask. Instead, I wrap around my face (leaving a slant gap beneath my hat for my eyes) the orange and yellow scarf I knitted as last winter's hobby. It is bright. And big. I could see how it could be attention-grabbing, especially when wrapped around my face. For whatever mysterious reasons, however, I am convinced that it is less conspicuous than the simple black facemask that I bought from REI.
As for doubling my gloves, I cover my wind-proof gloves with the double-layered woolen mittens that I nabbed from my little brother (6'4) when he was visiting last year. Because his hands are double the size of mine, they allow ample room for the insertion of activated hand-warmers.