Testnet for bitcoin - are there multiple operating?
There is no such thing as a private testnet, you can obviously create one, but then it won't be of much use to provide data regarding a private blockchain to the public.
The testing network (testnet3 presently) is the same for every user/developer.
BitEasy's block explorer for the testnet doesn't seem to be working. They don't have data even for dates a month prior to the date you asked this question.
As far as exploring the blockchain the data is concerned, you can do it yourself by accessing the block-chain on your machine after running the testnet bitcoind. The approach may be vary, but then the point being you can do it yourself if you know basic programming.