In my area for horses that are not particularly "special" in any way (not trained to high level competition with an excellent show record, for example) a half lease is generally half the cost of the horse's regular care- when I half leased a horse out, I calculated how much his regular vet, farrier, and board was for the year, then divided by 12 to determine his monthly upkeep, then charged half of that. The person leasing the horse paid directly to me, and I paid the farrier/vet/board bill.
For full lease/off-farm lease, you'd generally pay all of the horse's regular vet, farrier, and board fees directly to the provider. You might be required to use the horse's regular vet & farrier, so you might pay more than if you were able to choose your own.
As for actual dollar figures, it's impossible to say. In my area an average half lease could be anywhere from $150 to $400+ It depends on how much the board is where the horse is kept, whether he needs a full set of shoes or goes barefoot, etc. Some contracts may require you to take lessons during the course of the lease. In areas where horse keeping is generally more expensive, $400 might be the cheapest half lease you could find.