Are cryptographies based on bitwise operations considered secure?
If I understood correctly (using uppercase notation for public keys and $\lor$ instead of "|" to indicate the logical or) your scheme is:


Public Domain $g \in \mathbb{Z}$
Alice Private Key $a \in \mathbb{Z}$
Alice Public Key $A=a \lor g$
Bob Private Key $b \in \mathbb{Z}$
Bob Public Key $B=b \lor g$


Shared Secret = $a \lor B = b \lor A = a \lor b \lor g$ 

Then I would say your system is insecure as an attacker can recover the shared secret using only the public keys as:

Shared Secret = $A \lor B = (a \lor g) \lor (b \lor g) = a \lor b \lor g$
(Given the property that $x \lor x = x$)