When designing for print, having a properly calibrated monitor makes all the sense in the world. If done perfectly, what you see on the monitor should look like what you get on paper from a printer. No surprises.
When designing for the web, there is no physical byproduct in which to match color. We designed it on a screen, for a screen. So what does calibration matter?
One school of thought is definitely for it and suggests fancy devices. They might also suggest that there is more to calibration than screen-to-paper. Calibration is all about adhering to a known standard. So if you worked in a team and you all had well calibrated screens, you could be sure what you are looking at is what they are looking at.
Another school of thought might be to purposely have crappy calibration. After all, the vast majority of people visiting just about any website can't be counted on to have nice calibration, so if your site looks good and readable even with garbage calibration, it should be good for everyone.
I haven't spent much time thinking about this, as it's never come up as a problem for me. I think it's interesting though, as there is apparently some strong opinions about it. If you have one, I'd love to hear it.
Thanks to Clay Doss for sending in the question.