The mechanics are worth laying out plainly, because they explain why the outcomes vary so much.
When a borrower defaults, the bank or servicer already owns a problem. They have a loan they can't collect on, capital tied up against it, and a foreclosure process that costs them money and takes months to years depending on the state. Many of them would rather sell the debt at a discount and move on than run the process themselves. That's where a note buyer comes in. You didn't cause the default and you can't undo it, but you now hold the decision about what happens next.
A note buyer who paid 55 cents on the dollar has room a bank at full basis doesn't have. They can accept a reduced payoff, cut a payment down, or forgive some arrears and still make money. A servicer holding at par often can't do those things without a loss, so they foreclose. That's the real reason a workout can leave a borrower in the house.
The other side is honest too. Some buyers are collateral buyers. They want the house, they underwrite the foreclosure timeline, and any conversation with the borrower is about getting them out efficiently. Both types are in the market and both are legal.
One thing that shapes this more than most beginners expect: the rules on how you can contact and treat a borrower on a defaulted consumer mortgage are extensive, they differ by state, and some states require a license to acquire or service residential mortgage debt. That's a question for a lawyer in the state before you buy anything, not after.