Back to the first answer. It is the dealer's choice. He/she can conduct business anyway he/she sees fit as long as it is legal.
However, when you are told that it is illegal for an individual to ship a handgun across state lines or that the FFL does not have to send a copy of his license to an individual, you are dealing with someone who is either ignorant of the law or is lying to you "for business purposes". I would not trust that person to do my handgun transactions. If he/she tells you they are unwilling because they are uncomfortable or he/she believes it will be easier to resolve a problem if it is FFL to FFL, then at least they are being straight. In that case, it is your choice. If you are the seller and are lucky enough to have an FFL who will do the transfer and shipping reasonably, it is a no brainer.
Being an FFL dealer for 20+ years, I have to disagree with the hi-lighted statement. It is absolutely illegal for unlicensed individuals to ship a hangun across state lines UNLESS it is being shipped to a bona fide FFL or licensed manufacturer, and there is no requirement whatsoever for a FFL dealer to supply FFL copies to unlicensed individuals. On the other hand, if you meant to say a FFL dealer is telling you he cannot ship your gun for those reasons, then yes, he is full of manure.
I have no qualms about giving copies of my FFL to established dealers or people known to me. Most of the above posts that speak of dealers freely sharing their FFL, indicate they are either frequent customers, friends, or known well enough by the dealer to be trusted. FFL copies can be modified, recopied, photoshopped, etc, to show anyones name and address and match any ID. This is one of those things we hear about but don't have a specific incident to point out....the kind of thing someones brother in laws cousin heard about from his nephew but can't recall when and where....

Anyway, in this day and age of computerized counterfeiting, it would be a piece of cake to turn a legitimate FFL into a bogus FFL That's why I would rather supply the FFL numbers to the individual....that is all anyone needs to verify the FFL dealers correct name and address via the ATF web site or a phone call to ATF. There is no need for the unlicensed individual to have a hard copy of anyones FFL...as already said..."what would he do with it?"....there is nothing legal he can do with it, but there is a possibility of illegal use. Some individuals send me a copy of their drivers license, or concealed carry permit, and some don't. All I need is their name and address for my records and I'm happy. Keep it simple and uncomplicated.