I don't list my current employer on FB. Ever. I will list past employers, though. But people can't search for me unless they know me already. It can't be done. I've had too many scares with actual crazy people threatening me. Sorry, no.
The only person from my current company I have on FB is my mom, and I think the personal connection between us kind of blurs the lines of "fraternization" between manager and employee. (Note, she's not a direct manager above me, she's in an entirely different department.)
My company itself doesn't have a FB page. Our parent, which is also one of our biggest clients, does. We're encouraged to follow and like them, and share their posts, but I don't. If I say something about work, it's completely vague, so unless someone knew the situation, I could be talking about my job or working in the yard or working around the house.
At my old employer, we ended up firing three employees and writing up five more because of FB. The write-ups were completely ridiculous, in the opinion of everyone up to and including the distrcit manager, but it was policy, you see. People who had been FB friends for years, since they all signed up, were told to un-friend one of them because she had recently gotten a promotion. Technically they did, but they still wrote on each other's walls and left comments because all of their families were still friends. And because they had interaction outside of work as equals, that was against the company policy. They were discussing pregnancy, children, family outings, cute things their kids said, nothing at all that could possibly be construed as detrimental to the company. So where is the line for FB use and how much say an employer has in personal use?