Is this method for exchanging private key using RSA sound?
You are inventing your own protocol.  Don't do that!  You're likely to get something wrong.

Instead, use SSL.  You can use SSL with a server cert and a client cert.  That solves exactly this problem; each party can verify the public key of the other end of the communication and verify they're talking to who they want to talk to.  If everything checks out, they have a secure connection they can use for their communication.  It solves the problem, and it has been vetted far more thoroughly than anything you invent yourself ever will be.