Even at larger companies this is a problem, if you're not part of huge team all with similar skills and specialties. The company I work for has ~3000 employees nationally, but if you combine geographic location, education, the type of work I do and skill set, you'd be be able to pick me out from those 3000 fairly quickly.

When you sign up, they have a question "does your company have less than 10 employees", which is presumably to deal with this issue. Of course, 10 is a pretty low cutoff for this question.

Yep. We hide the company name if you mark this as "Yes".

Really. I can't think of single time when knowing where a candidate is/was working influenced me or any other interviewer on my team. It simply doesn't matter.

OK, there were a few times when we said in a debrief "how did someone so clueless ever get hired at XXX" but that's hardly a positive influence :-)

Yup, exactly. Decent idea, but this kills it completely.