How is the key in a private key encryption protocol exchanged?
Whenever Alice and Bob wants to agree on a same private key, the most popular method is to use Diffie-Hellman. It works as follows:


Two public values are first chosen lets say $n=13$ and $g=17$.
(These are usually very large prime numbers and known to everyone
using that protocol).
Alice chooses a Private Value $a = 3$ and Bob chooses a Private
Value $b = 7$. These are private to themselves.
Alice computes: $A = g^a\mod n$ and Bob computes: $B = g^b\mod n$,
in this case $A = 12$ and $B = 4$ and they exchange each others
values (may be through chat), i.e. everybody knows the values of $A$
and $B$.
Alice computes: $K = B^a \mod n$ and Bob computes: $K = A^b\mod n$, in this case $K = 12$.


Now Alice and Bob have both agreed upon the value $K$ as their key. Note that since the value $n$ and $g$ and very large prime numbers it is almost impossible for a eavesdropper to factorize them and compute the key himself.

A problem in private key cryptography is man-in-the-middle attack and that is one of the main reasons to choose public key cryptography over private key cryptography.