@Profcreep To make it simple the way mod menus are written makes it so that it identifies the host as the person hosting. When you play a private game you're both the client and server so you are host and get the mod menu. When you host a dedicated server your client (you) is a random person and the server is the host so the dedicated is the one who has the mod menu (so in the end no one)
To make it work for you either host a custom game or like mentioned above change the GSC script to give host status to you.
Here is a topic that explains it https://forum.plutonium.pw/topic/1212/release-bo2-gsc-gr3zz-v4-1-zombie-mod-menu