We've got both the Nikon 12-24 and the Sigma 10-20. I bought the 12-24 first, but had to check my bag each time I left on a shoot to make sure my wife hadn't lifted it for her own shoots. I finally bought the 10-20 out of self-defense. Now she's checking her bag to make sure I haven't swapped out HER 10-20 for MY 12-24.
The Sigma may cost less than half the Nikon, but in the particular examples we own and shoot daily in our business, the Sigma outperforms the Nikon on all counts. You can sort the images by which lens they were taken with. The Sigma is sharper, more contrasty, and has better color saturation. On top of all that, those silly little 2mm of extra coverage at the low end are really significant. When you need wide coverage, you really need it and the Sigma is a lot wider.
Buy the Sigma 10-20 and don't look back.