Here's what I think a good game designer should do:
1. Communications - IMPORTANT. Able to articulate well what you think and your ideas to others, both verbally and written. Your job is not just about expressing your own ideas, but also hearing other people's ideas, from developers to QA to managers to other people.
2. Drop the ego - Many self-proclaimed 'game designers' think that their idea is the best and everybody should listen to his idea. You don't make games for yourselves. You make games for others to play. If other people don't like your idea, then admit it. Alternatively, other people may end up with better ideas than yours, in which case, you should be willing to adopt this new idea and flesh it out.
3. Creativity - Be creative, but not too creative. Although your idea may be good, but being too creative can make it complicated, and sometimes people don't want complicated mechanics in their game. Again, listen to feedback, back to point #1.
4. Have other skills under your belt, programming, but artistic skills is preferred as it can help you in communicating your ideas better (see #1 above). Many designers that can draw/sketch and design levels would sometimes do so to get their point across.
In the professional world, you will always work with other people. Being a game designer does not mean that you have other people to create your ideas for you, and all you do is just sit there and coming up with ideas. You become the hub of communications between engineers, artists, testers, producers/managers, and other stakeholders. Your job would mostly be answering emails, making phone calls, attending meetings, and making sure that the game stays on track when it comes to the whole look and feel. Engineers can easily lose track on how the game looks as they are concerned about getting things to work and fixing bugs. Artists can easily lose track in the feel of the game as a whole as they tend to narrow their art on parts of the game only (i.e just making those textures). Testers are your good feedback generators as they will be the ones who would genuinely tell you if game sucks or not. Then managers only care about money, so they tend to be stuffing down everyone's throat to make your game just like Farmville.
So you sit there, in the middle of all that chaos, and making sure that the game is still fun while people are complaining left and right about it. As you can see, at the end of all that process, you might only be able to squeeze in 10% of your own original ideas, while the 90% are others.