Scenario A: You find a house for 100K that's turnkey ready, and will generate 12K/year in net cash flow after all expenses except the mortage. The bank wants 25% down, so you have 25K invested, and then you make whatever's left over from the 12K after you pay the mortgage (let's say it's 6K left). And the bank will take 30 days to close, they require a credit report, etc.
Instead, you go to your friend John, who's a doctor and has a lot of cash but not a lot of time. You convince John to give you the 100K to buy the house (you put in no money). You'll deal with finding the house, placing tenants, managing the property, etc. In exchange, you agree to split the profit 50-50. So of the 12K, John gets 6K, you get 6K. He's still making a 6% return, and he didn't have to do anything. You're making 6K for finding and managing the deal, and you didn't have to put any of your own money in. Alternately, maybe John would rather have a guaranteed rate of return instead of equity - so you tell him you'll give him 7% interest on his 100K, plus a first-position mortgage on the property (so he can foreclose on you if you stop paying him). Now, you pay John his guaranteed 7K/year out of your 12K profit (just like you would pay a bank), and you keep 5K. If you have a bad year and only make 10K, John still gets his 7K, and you only keep 3K. You have a great year and make 15K, John still gets his 7K, you get 8K.
Scenario B: You find a distressed house that all fixed up would be worth 110K. It will cost you 50K to purchase, and 25K to renovate, so 75K all-in.
You go to the bank. The bank says "it's only worth 50K right now, so we'll only give you 75% of that, or 37.5K, and you're on your own for the rehab. We'll charge you 6% interest, and we'll take 30 days to close the loan." So now you have to come up with the other 37.5K on your own to get the entire 75K you need to do the deal.
Instead, you go to John and say "give me 75K for a year. I'll pay you 10% interest." John gives you the 75K the next day, so you go buy the house and rehab it. Now it's worth 110K. You wait a year until the bank will let you do a cash-out refinance (i.e. the bank's "seasoning" requirement). Now they'll give you 75% of 110K because it's all fixed up, or 82.5K.
Now, you pay off the loan from John, plus interest (75K + 7.5K = 82.5K), with the money you got from the bank. Now, John has his original money back, plus a year's worth of really good interest (10%), you bought the property with zero of your own money, and you have a (hopefully) cash-flowing rental that easily covers the new bank mortgage, and you still have 25% equity left in the property.
Again - you can structure a private deal any way that makes sense to both you and the person lending you the money. Some people want a share of the profits, some are happy with a fixed interest rate loan, some want points or a down payment up front, some want to fund the entire deal and are happy getting a lump sum check at the end.