PS: How is this method exacly called that one player hosts game for others? It's not peer to peer I guess and not simply client/server?
It's just client / server. If you want secure games, you will have to host on machines you trust. Either because you own them, or you can validate and authenticate the server content. Usually 'pure' game servers are called dedicated servers. There is no client-side, no rendering or sound, and the game is controlled via a console (entering commands, either locally, or by a games administrator remotely, rcon_password and all that). It's a lot lighter on resources, and you can run several servers on one machine, your bandwidth becoming your bottleneck.
Internet security is a big deal, especially when money is involved (auction houses). Then you run into legal matters. Simply put, a client / server would be the most secure you can hope for, and from where you can build on, and if someone wants to run a moded server and you do not run an authentication service, then there is nothing you can do, apart from accepting it.
If you run a lobby service, then you can run an authentication service. You will need to authenticate users, client and server content, client and server binaries, transactions, communications, ect... SteamWorks, XBox Live or Games For Windows Live are examples of such systems, and as you can tell, it's not an easy thing to do.