"How many English words does an English speaker know?"
How long is a piece of string? There are all sorts of speakers out there. You have to be more specific.
As Paule89 mentioned, a quick Google search will reveal some very interesting results. According to some research (I'll have to look up the source when I get home), native speakers don't use much more than 5,000/6,000 words (or rather word families) on a daily basis. That is, allegedly, enough to understand 98% of what you usually hear or read.