I think some of the obscurity about birdseye grain comes from the fact that, in regular lumber, "birdseye" is a figure that occurs when you have a lot of tiny knots widely distributed over a board, most commonly in "birdseye maple." In briar, however, what we call "birdseye" is a pretty normal property of the burl from which the pipes are made.
If you look closely at just about any smooth briar pipe, you'll probably see birdseye figure somewhere, depending on how the original burl was cut. In a pipe made from "ebuchon" briar, you'll often find the birdseye figure on the sides, though it will probably be more prominent on one side than on the other. In a pipe made from "plateaux" briar, you'll find the birdseye on the top or bottom, and often both. In general, blocks cut from the top, or the outside of the burl tend to be cut as plateaux blocks, whereas pipes cut from the inside tend to be cut as ebuchon blocks. But what normally distinguishes ebuchon from plateaux is the orientation of the grain. I'm oversimplifying a little bit here, as briar cutting is an art in itself, but that's the gist of it.
So, for example, here's a pipe I made some time ago (yeah, I know, not a great looking one, but I'm still a novice) from ebuchon briar, and you can see a bit of birdseye on the side:
Here's one I made from plateaux:
Notice the vertical lines in the grain. The lines terminate in the little bumps on the top of the pipe. If you remove that natural surface, you reveal the birdseye grain underneath.
Because we're dealing with a natural substance here, there is a lot of variation in the appearance of the birds' eyes from one block of wood to another. Naturally, the ones with tighter, more vivid figure sell for a premium and tend to find their way into pipes that also sell for a premium.