I started out half leasing a lesson horse from my barn. They had their half lease fee set at half of board (board at the time was $325), which would give you 3 days a week to ride. I was fortunate in that the horse I leased was only used in 1 or 2 lessons each week and I ended up being able to ride whenever I wanted, so it was kind of like getting a full lease for half price
It was a month-to-month contract, so I could cancel any time with 30 days notice. IME, private owners looking to half lease out their horse tend to factor in other costs like farrier, annual vaccinations, etc. so they often charge a little bit more than half the boarding costs, or may charge a flat monthly fee roughly equivalent to half board and then expect you to pay half of farrier costs as they come up. Private owners may or may not want a long term commitment; it often depends on why they're offering to half lease out their horse (for example, if they're offering a half leasing while away for college, odds are they'll want a longer term contract so they have guaranteed income to pay for the horse's expenses while they're in school)
Before I took the dive into buying my own horse, I did a full lease of a privately owned horse (found on Craigslist). It was a "free" lease (that is, I didn't pay the owner anything), and I was responsible for paying the barn, farrier, vet, etc. directly for everything. In a lease situation like this, it's as expensive as you choose for it to be, since you've chosen the barn to board at, what farrier to use, etc. The owner may require certain things that affect cost, like using a specific farrier.
In general, expect to pay more if the horse has a good competition record and is show-ready. Also expect to pay more if the horse is being kept at a fancy barn.