Hi Horsef, All.
My animals are routinely both led, and mounted from the off-side (right side), so they have learned not to care which side I'm on. If I have to lead them _beside_ a road with much traffic, I will walk facing oncoming traffic, with the animal on the outside. In the USofA, this translates to walking on the left side of the road, and leading on the right side of the horse.
If there is a shoulder, I will walk as far off of the road as possible; right up against the fence line if there is one, even if it means wading thru weeds and such.
I really try not to lead, (or ride) _on_ a busy road. If I do have to get on the road (say to cross a bridge or something), honestly, I will go right down the middle of the lane, so drivers are forced to acknowledge me (and either wait, or pass appropriately), and then get off the road as soon as possible.
IMO, this is about the most unsafe thing you and your horse can do together, so please Be Careful.
Steve Jernigan KG0MB
University of Colorado