I've hired my sister as a graphical designer several times and been mostly happy with her work. She did the "Sherlock Owl" logo for my security lab (based on a rough idea I had going in) and the five-point-star-check logo thing for our voting research center. Her process is that she first talks to you about what you're looking for, then sends you a PDF with several pages of mock-ups. You then pick the ones you like, say what you like about them, and/or how you'd recombine them. Then repeat at least once, maybe twice. My sister doesn't do web design, though, so I cobbled that together and it was good enough.

I'm not saying you should hire my sister (although she's pretty good at this sort of thing), but that this is the sort of process you should be looking to do with whomever you end up hiring.