Saxon Pig is correct. The shipment must be TO an FFL holder (Dealer License or, if applicable, Collectors License), but shipment can be FROM anyone. But yes, always have a copy of the receiver's FFL before shipping and ship ONLY to that address. You also can check a dealer's license on the BATF web site to be doubly sure, as FFL's have been counterfeited. Then keep the copy for your records, just in case the shipment goes astray or winds up in the wrong hands.
True, some dealers will not send an FFL copy to a non-dealer; that is their choice and there is no law that will make them do it. But if you have good relations with your local dealer, you can often, as Saxon Pig says, even get extra copies of his license to give to a seller at a gun show or to send with an order.
One important point: "Shipment" of a rifle or shotgun across a state line can be by mail (US Postal Service), by anyone, as long as the addressee is an appropriate FF licensee. But handguns MAY NOT be mailed (USPS) BY anyone except a licensed dealer or licensed manufacturer. A Collector license (C&R license) holder may not mail a handgun to anyone, even to a licensed dealer or licensed collector. This is a postal regulation, not a BATF ruling. Some people evade this, and say that it is OK to mail handguns if the post office clerk does not ask what is in the package. Contrary to some silly ideas, postal regulations really do have the force of Federal law and there is no "don't ask, don't tell" policy; if the package is x-rayed or comes open, the sender could be in for a serious fine or jail time.
Jim