I very well understand the DH key exchange. But what is the public key and private key at the end of the exchange.
For example: Alice and Bob agree to use a modulus $p = 23$ and base $g = 5$ (which is a primitive root modulo $23$).
Alice chooses a secret integer $a = 6$, then sends Bob $A = g^a \bmod p = 5^6 \bmod 23 = 8$
Bob chooses a secret integer $b = 15$, then sends Alice $B = g^b \bmod p = 5^{15} \bmod 23 = 19$
Alice computes $s = B^a \bmod p = 19^6 \bmod 23 = 2$
Bob computes $s = A^b \bmod p = 8^{15} \bmod 23 = 2$
Alice and Bob now share a secret (the number $2$).
So is $2$ the private key here ? If it's a private key then both Alice and Bob know it even though the eavesdroppers don't know. Please explain how public key/private key pair is generated from this shared secret $2$.