Not sure I understand, so let me do a couple of examples. I will assume you will hire out all the work and that neither partner puts any value on the other's work of finding the property, hiring contracts of other such activities. The only that matters is the money.
Say the two of you buy a house for $50K that needs $20K of work. You'll have some closing costs when you buy, maybe $1000. You'll have a bunch of costs when you sell, maybe $10,000. You'll also have insurance, taxes, utilities and other holding costs. Lets guess those at $1000. So, the deal looks like this:
Selling Price: $100,000
Less all these costs:
Sales closing costs: $10,000
Buy closing costs: $1,000
Holding costs: $1,000
Purchase price: $50,000
Fixup costs: $20,000
Total costs: $82,000
Now, you don't actually need $82,000 to do this deal because the $10,000 in sales closing costs come out of the sales proceeds. So, you actually need $72,000 to do the deal.
The simplest thing to do would be for each of you to put in $36,000. Each of you does half the work involved in making the deal happen, or, perhaps neither of you does enough work that the other cares. When you sell, you get $90,000 back from the title company after the closing costs. First, each of you takes back your $36,000 investment. That leaves the $18,000 profit. You split that 50/50 and each gets $9,000.
The "money partner" situation. Here one partner puts in all the cash and the other does ALL the work. Again, we'll do a 50/50 profit split. The money partner puts in $72,000. From the $90,000 you get from the title company, the money partner first gets $72,000. That leaves $18,000 profit. The two of you split that 50/50, giving each $9,000
Now, what if one of you puts in cash and the other gets a loan? It looks a lot like the first case. One person puts in $36,000. The other person get a $36,000 loan, secured by the house. When you buy, you need $51,000 to cover the purchase and closing costs. So the person putting in the loan signs the loan paperwork and the other person brings a check for $15,000 ($36000 + $15,000 = $51,000). The money partner pays for the repairs and holding costs for a total of $21,000. That means they've put a total of $36,000 in.
The person who took out the loan has to make the payments. That money does not count for anything.
Closing works differently this time. In this case, the same $10,000 of closing costs are subtracted from the $100,000 sales price. But the $36,000 loan also has to be paid off. So, the check from the title company is only $54,000. That's because the loan's already paid off. Out of the $54,000, the money partner gets $36,000. That's their investment. That leaves, ta-da! $18,000. That's your profit, which you split 50/50.
Now, why is 50/50 not a fair split? After all, you each put in half the money. Its not fair because the property was used as collateral for the loan. If the person taking the loan used something else for collateral, such as their residence, then I would say a 50/50 split was correct. But if the property itself serves as collateral then the borrower's risk is MUCH less than the money partner's risk. In fact, the borrower's risk is almost zero.
What? Think about what happens if something goes wrong. You buy the house, fix it up and nobody buys it. Months and months go by, still no buyer. Say the borrower gets tired of paying on the loan and stops paying. Eventually the lender forecloses and takes the house back. By that point they're owed $45K between the loan balance, missed payments and fees. The house is a mess because its been neglected for months and months. But you did put some work into it. It sells at auction for $55K. They take their $45K and hand you back the $10K that's left. That goes to the money guy. His loss is $26,000.
Or, nobody buys it at auction, and it goes to the bank. The money guy has lost $36,000. The borrow has a big hit on their credit, but loses nothing.
Or, say you find a buyer, but at $75K not $100K. After sales costs you get $67,500. (I just repossessed a house with number very much like this, bigger numbers, but the same ratio.) $36,000 goes to pay off the loan. That leaves $31,500. That all goes to the money guy, who's owed $36,000. The money guy takes a $4,500 loss while the borrower gets off with no loss at all.
What's the right split in this case? Beats me. To be honest, a deal like this is to complex and has too many chances for something to go wrong to even try to do, IMHO.