There's pros and cons to both.
If you ship your own bike, it's kitted out the way you like it: seat and shield tailored to your comfort, you know what will or won't fit in your panniers and cases. Plus you can ship all your motorcycle gear with the motorcycle, and walk on the plane with nothing but a backpack. If I was taking a multi-week motorcycle trip, I wouldn't want to sit on the stock seat of a rental bike every single day, or have to leave gear behind in the rental company's office because the luggage options were too small.
However, if you rent, you don't have to worry about insurance - which is something you have to shop for and purchase separately when you ship your own bike. Also, it's much cheaper for shorter trips, and most reputable rental agencies carry late-model vehicles in good condition, so maybe you get to try out a nice BMW or Ducati.
Most companies also have a breakdown service included, so if your battery goes or if you get a flat, they'll send someone to fix it. Saves a lot of time and money especially if your vacation is short - you don't want to be calling around for a shop and/or a tow truck company when you could be back on the road in a few hours.
Once, a rental company brought a new bike to replace the one we were riding because it broke down. The guy rode up to us, handed us his keys and loaded the old bike up in a van that was following him. All within 45 minutes of us calling the rental agency. That was pretty sweet.
Another problem with shipping is that sometimes your motorcycle won't arrive on time, depending on airline delays or administrative SNAFUs. If you've only got 10 days vacation and your motorcycle arrives 7 days late, that would really suck. That happened to us a few times. Fortunately once, it was at the end of our trip and the bikes were being flown back from Europe to Toronto. Somehow they overshot and ended up in Vancouver. It was cheaper for the company to truck them back to TO, and we got the bikes back a week later.
No apologies, no compensation. That's just the way the airlines roll.
Pros and cons.