It can't use its private key since anyone would be able to decrypt the message, so my question is: Does the client have a public key?
Please note that the public / private key is (by default simple SSL) used only to authenticate the server, client can stay anonymous. The server's public key is used only to safely exchange (encrypt/decrypt) random session keys which are used for symmetric encryption (see https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange + server can sign the generated session key with its public key to ensure it's identity )
These symmetric keys are used to encrypt communication between the server and client (both way), so the client doesn't need any public key (or certificate) to establish secure communication.
The client may have its own keypair (and certificate), they are used to authenticate the client, not decrypt the messages.