Why is a proxy needed in proxy re-encryption?
If Bob has the re-encryption key from Alice, then he can re-encrypt all Alice's ciphertexts stored online and then decrypt these messages. This is not good because Alice may only want to reveal some small subset of messages to Bob but not all. Similar things happen if Alice knows Bob's re-encryption key. Furthermore, the proxy is also not fully trusted, i.e., knowing Alice and Bob's re-encryption keys is not enough for the proxy to decrypt the online ciphertexts either (because it does not know Alice and Bob's private keys). On the other hand, if the proxy and, say, Bob collude, then they can decrypt all Alice's ciphertexts.