In this work we study the spectral functions of scalar mesons in one- and two-channel cases by using nonlocal interaction Lagrangian(s). When the propagators satisfy the Källen-Lehman representation, a normalized spectral function is obtained, allowing one to take into account finite-width effects in the evaluation of decay rates. In the one-channel case, suitable to the light σ and k mesons, the spectral function can deviate consistently from a Breit-Wigner shape. In the two-channel case with one subthreshold channel, the evaluated spectral function is well approximated by a Flatté distribution; when applying the study to the a0(980) and f0(980) mesons, the tree-level forbidden KK decay is analyzed.