I like making game concepts for fun, as a hobby. I don't think that all my game concepts are amazingly awesome and have to be created and would revolutionize the industry. I just want some people to talk to that have the same hobbies as I do. To be able to discuss ideas and point out the pros and cons.
Hobbyists, independent developers, and prospective developers discuss these things on this site, Gamedev.net, and various forums such as TigSource and Indiegamer. These are pretty much only the places where this sort of discussion happens.
By comparison, professionals tend to carry out their high-level discussion of game design in a similar way to much academia: they publish books, papers, articles, give conference talks, and post blog entries, which are then discussed and cross-referenced. There's no reason why you couldn't start your own blog about your ideas, but you'll have trouble getting any traffic, because the average person's game concept is not going to be of much interest to the wider community. This is why typically you'd go for the first route, of game development forums.
The reason there isn't anything more for this is partly because the professionals do a decent enough job of exchanging and discussing ideas, and partly because the ideas and concepts that a typical prospective developer comes up with are considered nearly worthless in the professional arena. (And as such, only of interest to one's peers on forums and the like.) This perception of worthlessness is because most professional game developers already have more of their own pet ideas than they have time to create (making ideas far too abundant for the average one to be worth much), and because the quality of a game is almost never intrinsic to the concept itself, meaning it can be almost useless in isolation without a full prototype or technical specification behind it - and usually the only people capable of making such a prototype or specification are already either in the industry, or are able to make the game themselves without presenting it to others.
There's a design board on the TIGSource forums. It seems pretty full and diverse.
You might try the GDSE Chat Room. It's occasionally pretty well-populated, and has proven to be a pretty good sounding-board.
I have a website myself that I am using for game design discussion called Ascension Protocol. You can visit it at: http://www.ascensionprotocol.com, be sure to visit us on facebook as well!