I leased several horses before I bought one and have myself now leased our horses out for many years. Most folks will do either a full or half lease...some will even consider a quarter lease. It is fairly typical to charge for board and farrier fees. Some owners may also charge for regular, predicatable costs such as supplements, worming, etc. So if you do a full lease, you'd pay 100% of those agreed upon costs and have access to AND be expected to care for the horse 7 days a week, if you did a half lease, you'd be expected to pay 50% of the those agreed upon costs and have access to the horse 3-4 days a week---most owners will ask you to commit to specific days, like Mon, Wed, Sat or something like that, but it depends. Something that is important to note is that typically, if you lease a horse, you are expected to care for that horse on those days, i.e. it's not just a "oh I might go and ride on those days" it's a "you are responsible for the well-being and care of that horse on those days."
In my opinion, no lease should require you to pay unexpected veterinary costs---just too much trouble can pop up from that. ALWAYS get your lease contract in writing and signed. This protects both you, the owner, and the horse, and is also a good way to clarify rules and expectations.
To me, leasing is just fabulous! It gives you a taste of horse ownership without the true committment and financial risk and burden of owning a horse.