Summarize:

They cannot be earned in any way unless you have been playing Animal Jam for a long time. Or if you are a member, with an Eagle, Owl or Falcon, you can play The Forgotten Desert to get the betas. Some betas are impossible to get from The Forgotten Desert. Some of the sought-after betas are:  Blue/Orange Mat Cami's frog Gecko Plushie Worn Blanket You will have people trading you rares constantly. Alternatively, look on people's trade lists for a beta and trade your best stuff for it. Go whichever way you prefer.
Note that betas are items that were sold during Beta Testing. Know the betas. If you get your hands on a beta, put it on trade to get other rares.