I'm not a professional, but I'll try to answer anyway.
Meteor showers occur when the Earth passes through the orbit of a comet (or, in at least one case, an asteroid). Over time, the debris spreads over the entire orbit of the comet.
A shower can last for several days, which is an indication of how wide the debris stream is. Assuming a duration of 1 day, and assuming the Earth's orbit is roughly at right angles to the debris stream, that gives a width of very roughly 2.5 million kilometers (and a length of several hundred million kilometers). The Earth is only about 12,735 kilometers in diameter.
Say the comet's orbit is 1 billion kilometers long (that's probably shorter than average). Then multiplying the length of the orbit by the area of a circle 2.5 million kilometers across gives the volume of the stream, and multiplying 2.5 million kilometers by the area of a circle 12,735 kilometers in diameter gives the volume of the stream through which the Earth passes (the hole it punches in the stream). The ratio is about 15 million.
- Earth's gravity will pull in some debris that wouldn't otherwise have hit it, making its effective diameter a bit bigger (thanks to ghoppe's comment).
- The density of the stream is not uniform. There are bound to be clumps of greater density. There's probably also a systematic change of density with distance from the Sun. The width of the stream probably varies as well. I have no idea of the details
- The Moon (and its gravity well) will also sweep up some debris -- but the Moon's effective area is a small fraction of Earth's.
But the blatant errors in my assumptions undoubtedly swamp any such effects, and I'm only looking for a rough estimate.
So yes, the Earth's passage through a meteor stream will effectively punch a hole in it, but it's a very small hole relative to the size of the entire stream. It could have a significant effect over millions of years.
I'm making a lot of simplifying assumptions here, but the conclusion seems about right if I've gotten the result within one or two orders of magnitude.
Reference: http://www.amsmeteors.org/meteor-showers/meteor-faq/#5, plus some of my own extremely rough back-of-the-envelope calculations.