I think part of the problem is that because the Internet is still a relatively new concept in the business world, companies often have no idea about websites. I don't know how they got the idea that just 1 person can do all of these marvels, but unfortunatly for us, some of them do.
If a company wants to be a success, they need a team of web designers, graphic designers, programmers and webmasters.
And if your in the market looking for a job, try to find a company that has a team, not just 1 or a couple of people working on a website. Be realistic, if you think you can't do 20 different requirements, make sure that you apply to a company that has the proper resources available to you. The last thing you want to do is overburden yourself and make you look bad to your employer.
I think the best thing to do would be to be honest. Tell employers what your strengths are, what you specialize in, ask them what they expect from you down to the letter.
I made the mistake of accepting a job (my current job) that relies totally on me for everything computer related. While it's nice being on my own sometimes, I now would much rather be a part of a team where I could gain experience from others. However I guess everyone has to learn that the hard way sometimes.