Segregation of Si and Mg at grain boundaries (GBs) in Al and Cu has been investigated using density-functional theory calculations combined with recently developed local-energy and local-stress schemes. The physics behind the impurity-segregation energy is effectively analyzed by the local-energy decomposition. For the 9 tilt and 5 twist GBs in Al and Cu, Si shows large segregation-energy gains only at tighter sites, where local configuration of remarkably short Si-Al or Si-Cu bonds with high charge densities of covalent-bonding features are formed, leading to the local-energy stabilization as the final-state effects. On the other hand, Mg shows large gains only at looser sites. For Mg in the Cu GBs, the formation of stable Mg-Cu bonds or Mg states at looser sites is the origin of the preferential segregation as the final-state effects. For Mg in the Al GBs, however, the local energies of Mg-Al bonds are not so stable at looser sites, while the instability of Al atoms at looser sites in pure GBs before substitution is the origin of the preferential segregation as the initial-state effects. The behaviors of Si and Mg in Al GBs are dominated by the difference in local sp bonding nature among Mg, Al and Si, while Si-Cu and Mg-Cu hybridization interactions dominate the behaviors of Si and Mg in Cu GBs.