How does it work if someone takes out a loan to buy my old car from me?

Time to send that buyer on their way.
How they finance the deal is not your problem, just so long as they have the cash to pay for it before they drive it away. If they pay by cheque the car stays with you until the cheque clears.
If they want a title/VIN search they're welcome to get one but THEY pay.
If they're not happy with this the deal is off.
Sounds like a scam to me, have they mentioned how much this alleged search is going to cost you? You end up with a big bill for the search and no sale.
Tell them to go forth and multiply.

No and no and no. If you need to pay someone to buy the car, and that's what you are doing if you pay for a vin search, then you're doing it wrong. If they want a search done, they can do it themselves. Cash only means exactly that. You get real money in hand and they get the documents.

Private party=cash only. For every ones safety.

The seller has a car for sale for $XXX. and upon getting that amount gives the buyer a clear, signed title, keys to the vehicle, and perhaps a bill of sale showing how much was paid.

Then you notify the DMV it is no longer yours, and who bought it, and you are done with it.

If the buyer is taking out a loan, and the lender wants anything from them, that isn't your concern. Don't contact the lender, just tell the interested buyer if he shows up with $XXX cash the car is his unless someone else beats him to it.

Find out the name of the loan company and call them. Actually the vin number of a car is no big secret. Anyone can see it through the windshield of the car.

Any title search is paid for by the BUYER. This goes for buying a home, an RV, motorcycle, or passenger vehicle like yours, buyer pays inspection fees, title search, carfax or any other fees. You only need to provide a bill of sale and the title in exchange for greenbacks, dollar bills - not a check, unless it is certified, and no money orders. Perform the money exchange at a bank - with a notary public as a witness to your signatures and payment, odometer reading, etc. If you need to know what to include on the bill of sale, contact your DMV, they can help you, and can help you understand your rights/responsibilities in a private auto sale.

If you take a personal check, it may be a bad check, written on a phoney account, closed account, insufficient funds, etc. Cash or cashier's check from a well known, local bank will make things much easier for you to cash or deposit and use immediately. If you deposit a bad check, not only do you lose the vehicle, because you signed the title, but you also have NSF fees and other bank charges on top of that.

Yup pretty standard.

If you want to get a car loan the company lending the money wants to know what vehicle the loan is secured on, and check that there isn't already a lien / it's stolen / it's not as described etc. Otherwise they could be exposed to all sorts of fraud and dodgy dealings.

So whether it's bought from a private party or a dealer, and whether you are borrowing the money from a dodgy title lender or a large commercial bank it's the same. They need to know what they are actually lending the money on, and check there aren't any problems with the title.

If you buy though a dealer you don't see this happen, the dealer and finance company do this behind the scenes as part of the finance approval process. But with no dealer, you have to provide the vehicle details yourself