They would store their egos at home when they left for work and they would all cooperate in the best interests of the client. Unfortunately, this is not a perfect world. Egos and reputations get involved. Oneupsmanship sometimes plays a part. Mutual respect may be lacking.
Arrogance is always a possibility.
Disagreement on treatment regimen alone does not make for a messy situation. It can often lead to healthy debate and ultimately to a better solution. Give your vet and farrier the opportunity to work out a solution between themselves before you step in to play peace maker. If the disagreement goes beyond healthy and informative discussion, you have a people problem as well as a professional problem. You have to approach it as both.