Maybe it varies from state to state, but my BO's liability would not cover me for anything if my horse were to get loose and cause a car accident, damage property, etc. They require all boarders to carry their own liability insurance up to some amount (don't remember exactly what it was). My existing homeowner's policy covers liability for my horse, though, so I didn't need to buy anything extra for this.
I do buy mortality/major medical for my horse, though. I only have one (it gets really pricey when you start thinking of covering multiple horses!) and while he wasn't super expensive, he also wasn't super cheap. Vet bills add up very quickly and I like knowing that if something bad were to happen I could make decisions based on the best possible outcome, not based on what it's going to cost.