I think it is wrong to call bitcoin a "private" currency.
On the contrary, imo bitcoin is much more of a public currency that any national currency is.
I think, bitcoin might be called as a private currency, is because the fact that bitcoin can give us an absolute access in our asset without barrier from the governements or financial institutions that often called the third party. But, in the other hand, i also agree with your opinion that bitcoin is much more like a public currency.