I presume you are flying from Arizona? Which airport, as this will be the limiting factor? Is there any reason you MUST fly United? There are other carriers that charge anywhere from $0-$150, although there is no way of figuring it out without going through each's policy that fly from whatever airport you are using. Remember, a $100 increase in base fare may well save you $400 in luggage.

I have recently heard anecdotal evidence that BikeFlights can get bikes to Italy for around $200, but that still isn't saving you money, and I haven't personally seen a quote that low. Most of the quotes I have tried in the past are more in the $500-600 each way to ship to Europe. FedEx retail rates are even higher, pushing four digits. For my money, at least internationally, flying with the bikes has been the cheapest option by a long shot.