First of all, there is ONE, very LIMITED occassion where FEDERAL LAW requires the shipper to notify the company that they are shipping a firearm. It is contained in 18 USC 922 (e):
(e) It shall be unlawful for any person knowingly to deliver or cause to be delivered to any common or contract carrier for transportation or shipment in interstate or foreign commerce, to persons other than licensed importers, licensed manufacturers, licensed dealers, or licensed collectors, any package or other container in which there is any firearm or ammunition without written notice to the carrier that such firearm or ammunition is being transported or shipped;
IF the firearm is going to ANYONE within the same state, OR if the firearm is going to a licensed manufacturer, dealer or collector in ANY state, notification that the shipment contains a firearm is NOT required by Federal law and no law is broken if the shipper does not notify. Company POLICIES, NOT LAW, require notification of the firearm shipment.
Most states' laws do not specify what are considered reasonable measures to ensure the firearm is not sold to a prohibited person. Most states (if not all) have laws mirroring Federal law, which say that you cannot KNOWINGLY or REASONABLY SUSPECT that the buyer is prohibited. So, basically, ask for a ID to verify state residence (in your case, Florida), verify minimum age (in your case, 18), ask them is there any reason you are prohibited from possesing a handgun such as felon, domestic violence conviction, restraining order? If you have no other reason to not believe them, have them sign a receipt for the gun and be on your way.