What I have is a fleece scarf. I wrap it around my head (over my mouth in the front, back of neck in the back) and tie the rest under my chin. The part that's over my mouth is under moderate tension, but my beard pushes it away, so my breath goes up and down over my chin (not through the material). I then have the upper edge rolled to tighten it so that air can't escape upwards and fog my glasses.

It does get a bit moist after prolonged biking, but not badly, and that's not really against my skin.

The main effect is that there's a little pocket of exhaled warm air, just enough to mix with the next inhalation and help warm it a bit.