it is still "private" and should always remain private. but private means you are the only person having access to it not anyone else. like having an account on Coinbase as your bitcoin wallet, you won't be having any access to private keys that way.

the way people claimed BCH was either depositing their bitcoin on exchanges like bittrex prior to the fork so that bittrex and others give them the BCH tokens automatically (this is risky though).
or they waited for the fork to happen, then downloaded one of the BCH wallets, imported their private keys in that and claimed their tokens.