How can SSL secure a two-way communication with only one key-pair?
That's not quite correct.  In SSL, two things happen:


First, a session key is negotiated using something like the Diffie-Hellman method.  That generates a shared session key but never transmits the key between parties.
Second, that session key is used in a normal symmetric encryption for the duration of the connection.


SSL does use public/private in one way, because an X509 certificate is used to identify at least one end of the connection.  Those certs are signed using an asymmetric key pair.