The Washington Post's take on this buzzworthy story is slightly deceptive in using Instagram as a lure; in reality, the situation has very little to do with social media.
Richard Prince is an artist who for 40 years has been at the forefront of (sometimes barely) repurposing images in his work. Controversially, he uses found images from advertising, for example, and presents them from a different perspective, taking full credit for the new use.
He's in the news because he has a new series of images taken directly from Instagram posts to which he has replied. Prince has blown up the entire screen (capturing a user's image, user name and all the comments--including his own--as part of the new work. They sell for nearly $100,000, which is why the Post's article has latched onto the idea that our Instagram photos don't really belong to us.
It's not about Instagram. If you create an image, you are the owner, unless you post it using a service that claims ownership of anyone availing themselves of the service. Coca-Cola can't use your Instagram photo. If a pesky blogger like me uses it, you'd also be well within your right to send a take-down notice. Publicly posting something does not constitute relinquishing any rights to it; that's not what the concept of public domain means.
Rather, the story is that anything you create can be commandeered by artists, with some limitations. Usually, the use is supposed to be somewhat transformative, but art is impossible to quantify in this way. Prince has created ... assembled? ... some brilliant work that truly removes items from one context and forces viewers to consider them in another; that's art, and that should be okay, even if it should also be understandable for an ad agency designer and the photographer he or she paid a standard fee to both be uncomfortable that their work could legally appear in another person's artistic output without their input and for no fee. (The article is primarily concerned with money, but the practice also means your work can be used to evoke opinions you disdain. There is a control issue there as much as a licensing issue. But people would be less up-in-arms if the new works were not for sale ... that's guaranteed.)