The decomposition of <Ŝ(2)> for a general wave function has been carried out in the framework of the Hilbert-space analysis. The one and two-center components fulfill all physical requirements imposed to date. An inherent ambiguity of the Hilbert-space decomposition of a two-electron quantity, in particular using a Mulliken-type scheme, is also discussed in detail. The formalism of effective atomic densities has allowed us to derive in a simple manner appropriate expressions for the decomposition of <Ŝ(2)> in the framework of Hilbert space analysis that are consistent with Mulliken population analysis and related quantities. Using a particular mapping we have derived the Hilbert-space expressions also in the framework of Löwdin population analysis in a straightforward manner. The numerical results obtained with the latter formalism have proved to be more robust and reliable.