It depends on what you mean by "private". The word private is not really associated with Eris as far as I can tell. Rather, the word "permissioned" is used instead, and therein lies all the difference and your answer.

A "private" blockchain might imply a blockchain that is not shared with anyone. Such blockchains would effectively amount to slow databases and nobody would have much need for them.

Eris' value proposition over traditional databases is simple: integrity through cryptographically signed history.

What's stopping twitter from editing my tweets and making it seem like I said something I didn't say? Little to nothing.

This is where a blockchain approach comes in. If twitter stored tweets in a blockchain that others could copy, then any modifications that twitter made to this chain would be caught.

Blockchains preserve the integrity of the data within a database. They prevent people from cooking the books. This is of extraordinary importance and value.

Comparison to git