After doing some research on the issue, I think your friend's claim has some validity. One article says that China does tend to buy up all the plastics, and since the shipping boats have a large enough capacity to do so, they buy plastics in bulk from Europe and US. (http://www.guardian.co.uk/society/2004/sep/20/environment.china) According to another article, China ends up buying huge amounts of plastic because it accepts unsorted waste and offers higher prices because processing it at home is considerably less expensive with their low wages. (http://www.praguepost.com/articles/2005/07/27/china-buys-plastic-waste-as-demand-soars.php)

Missing from your friend's story is that there are many good legitimate commercial uses for recycled plastics, such as making textiles. Polar fleece, for example, can be made entirely from recycled plastic bottles. The part your friend might have heard about burning plastic might come from the fact that, like many things in China, some smaller factories have little oversight and regulation, and therefore might burn plastic unofficially.

If this is true, we need to find a way to get China to reliably regulate its own manufacturing industry. And that is easier said than done.