I suspect a lot of it comes down to sheer luck and how well the bike was packed. Here is my experiences from memory;

-- I shipped the first tandem frame & fork (that I had built from scratch) from Omaha to Seattle circe 1985 when I moved for a new job. The moving company ended up sitting it aside for a while at their warehouse in Seatle where they then closed the big hydraulic warehouse doors on it. It was bent it into a nice V shape, including the 2-inch CroMo boom tube, the 1.75 inch CroMo downtube, the top tubes and the direct lateral tubes (all were actually Reynolds 531 downtubes). Insurance paid a few hundred for it but it was worth a lot more and a bit of the heart was lost also. I will note that not a single joint cracked (all brazed with brass -- some were hand crafted lugs and some were fillet brazed).
-- I shipped several tandems between 2001 and 2009 using Greyhound and those all went well for $85 to $100 each. No damages. They know and understand tandems it seems. I have heard that they will not accept one if not well boxed up which is smart.
-- I shipped one clean, but used steel Santana from California to the midwest for ~$185 in 2009. The seller chose UPS and had just dropped it off and let local UPS store pack and ship it. It was really poorly packed by them and it arrived with the box in tatters, plus a bent RD, lots of scratched up paint, a torn saddle, bent brake levers, and I think a pedal was also missing... Anyway UPS paid the claim of just under $1,000. I got new set of decals from Santana to go with the new paintjob and bent parts were easily replaced (I didn't do any upgrades). I had to fudge the tubing decals as no one seemed to have any for the getting. I am still riding it and haven't seen a need to improve on it.
- I shipped two (or maybe three) tandems by FEDEX over the years for about $150 or so each without incident and no damages. I remember that when I mentioned it was a tandem, they perked up, quit worrying about the box dimensions and were very helpful and ready to serve. They knew and understood the carriage.
- I ordered a new $250 aluminum tandem (w/free shipping) off ebay for a low-budget friend in late 2009 and UPS delivered it perfectly in 4 days with not even a dent in the box. Turned out to be a pretty good bike...attractive and quite ridable (they are about $299 now I believe) after I took a hour to tensioned the wheels....Probably a weakness on my part as I just can't pass on a bike to someone else with wobbly machine build wheels that pluck poorly.
- I sent a friend on his way from the midwest to NYC area with a steel CroMo tandem in his van. Once in NY he got rear-ended by some guy from NJ and the frame was bent a lot as well as the rear wheel and RD. I would guess his insurance would have covered it but I didn't hear one way or the other. So "carefully" driving it yourself isn't a 100% answer either it seems.
- Another friend related to me that in 2011 he purchased a slightly used Miargi (sp) tandem and picked it up with his pickup. On the way back, the wind caught the tarp he had tied over it and the tarp and bike went airborne together. He noted that the bike hit the road "wheels down" and then shot over into the grassy median strip when it cartwheeled to a stop. He told me that "honest; there was no damage at all" to the bike and that he loaded it back up, lashed in down this time, and continued home with it. He had a few beers in him at the time he gave me the update so I never fully bought the story exactly...

Ok enough experiental data for today. I think the summary will be "to pack it extra well and buy insurance for full value". Then if there is shipping damage, take lots of pictures looking down into the box with the bike still in place and take pictures of any part of the bike that is sticking out through the box.

Hope that helps to add to what others provide.

/K