When one person is holding a horse and a second is working on the horse in some way - veterinarian, farrier, grooming - they should both stand on the same side. This is especially important if it's a procedure to which the horse might object.
If the horse moves suddenly, the person holding the lead line will most likely keep the head turned toward her, which will automatically cause the horse's body to swing the other way. If the two people are standing on opposite sides, this will cause the horse to bump up against the person working on the horse. If they're on the same side, then the horse's body should swing away from both of them, reducing the potential for injury.
So, when your farrier or veterinarian or anyone working with your horse moves from one side to the other, you move with that person. You'll both be safer.
Although most horses are trained to accept being handled - and mounted - from both sides, if your horse is at all skittish about it, start working with him now to get him accustomed to being handled from both sides.