It's all negotiable.
Sometimes yes. Sometimes no.
What you're talking about (excluding assumable loans, which are rare) is a technique called "subject to." That means you're buying a home "subject to" the existing mortgage. Shel-lee is correct that such transactions can trigger the "due on sale" clause. It doesn't happen often, but it can and she's obviously seen it occur.
The basic mechanics are simple: The owner deeds the property to you. You, in turn, promise to make the mortgage payments. An owner would be willing to do this to get out of a major problem--falling behind on payments, upside down on the mortgage and not able to sell, etc.
Usually, the amount of money paid by the buyer to the seller is minimal, if anything at all. It's more likely to occur when there's some equity in the property. Maybe the house is worth $100,000 and the seller owes $50,000. Technically, it would make more financial sense for the seller to sell conventionally, but if the seller needs to get out quickly, that might not be possible. So the seller could deed the house to the buyer and ask for, say, $10,000.
It's far less likely to occur if there's no equity in the property. Assume the house is worth $100,000 but the seller owes $120,000. Why would a buyer want to pay $10,000 to acquire a property that's already $20,000 upside down? (Actually, there are a bunch of reasons when it could make sense. But they don't occur all that often.) In that case, the buyer likely gives the seller little or no money. Coming in and taking over the mortgage payments is enough to satisfy the seller.
So it really depends on the circumstances and how good you are at negotiating.
Hope that helps.