Hi Bill,
First of all, it is completely legal for the individual in Maine to ship it to your FFL himself. He does not need to go through an FFL on his end. HOWSOMEVER... Many FFLs don't like getting firearms from non-FFLs. Ask your local guy FIRST if he will accept the firearm. Also, be sure the guy in Maine includes a photocopy of his driver's license. This will allow your Ohio FFL to properly transfer the firearm to you.
As to how he ships it - that is completely up to you and him to work out. He can legally use UPS, FedEx or the US Postal Service. Just be sure to have him follow all shippers rules and regulations so there are no problems receiving it on your end.
Hope this helps.