Private key is never sent; it would defeat the purpose of public-key security. You can exchange information with anyone even if someone spies on your connection all the time. You send public key to your partner. The public key is used to encrypt and cannot decrypt, you keep your private key which can decrypt but not encrypt. So, you can only receive a secret message from the partner. Apparently, to send a message, you need to receive a public key from the partner, not sent a private one.
You need to invert your thinking to get it.
Still not clear? Find detailed explanation here:
http://en.wikipedia.org/wiki/Public-key_cryptography[
^].
Just track what Alice and Bob do.
—SA