We modeled paramagnetic Si dangling-bond defects in amorphous SiO2 using a generalized-gradient density-functional approach. By creating single oxygen vacancies in a periodic model of amorphous SiO2, we first generated several model structures in which the core of the defect consists of a threefold coordinated Si atom carrying a dangling bond. These model structures were then fully relaxed and the hyperfine parameters calculated. We found that the hyperfine parameters of such model defects, in both the neutral and positive charge states, reproduced those characteristic of the E', in accord with experimental observations for amorphous SiO2. By eliminating a second O atom in the nearest-neighbor shell of these defect centers, we then generated model defects in which the Si atom carrying the dangling bond forms bonds with two O atoms and one Si atom. In this defect, the spin density is found to delocalize over the Si-Si dimer bond, giving rise to two important hyperfine interactions. These properties match the characteristics of the hyperfine spectrum measured for the S center. Our results are complemented by the calculation of hyperfine interactions for small cluster models which serve the threefold purpose of comparing different electronic-structure schemes for the calculation of hyperfine interactions, estimating the size of core-polarization effects, and determining the reliability of cluster approximations used in the literature.