Ok I just did this and shipped the car to Zurich Switzerland. Absoulutely the car needs to be paid in full by the buyer, typically a wire transfer into your account, or if a payoff is involved on your end the funds could be wired to your lender. Call them and they will advise on this issue - maybe two wires if you have equity in your car.
The real issue for the buyer is the title must be in his name prior to the car leaving the USA and then for arrival into the buyers country of delivery. Your responsibility is just to, provide a "Bill of Sale", odometer statement typically on the title, sign the title to buyer and he and or his agent can take it from there. Shipping is his issue - you just need to be available to hand the car over to shipper - they will pick up from an agreed upon location and handle it from that point.
If the buyer does not send someone by, in the USA, to inspect your car I would be concerned. Do not release the car until paid in full NO MATTER WHAT THE ISSUE IS and the funds have cleared your bank.
The Buyer will have to pay shipping fees, potentially taxes in the USA, foreign taxes , duty, VAT etc.. in the foreign country of residence. Still with all the extra costs the price may still be good for the Buyer.
GO SLOW AND JUST BE CAREFUL WITH AN INTERNATIONAL TRANSACTION - MANY SCAM ARTISTS OUT IN THE WORLD.