Confidentiality of "partial private keys" in certificateless public key crypto
The best way to answer this is to look at the security model in that paper.  If the adversary is allowed to request the partial private key for the target identity (i.e. the one it uses in the left-right query during the attack), then partial-private-keys are not meant to be kept secret.  If the adversary is not allowed to do so, then they must remain confidential.

After scanning their model, it seems that all partial private keys are readily supplied to adversary via Extract-Partial-Private-Key-of-A queries. This means they don't need to be kept secret.

I've got to run now, but I'll edit later if I've gotten this wrong.