We derive the scattering length of composite bosons (cobosons) within the framework of the composite-boson many-body formalism that uses correlated-pair states as a basis instead of free-fermion states. The integral equation constructed from this physically relevant basis makes transparent the role of fermion exchange in the coboson-coboson effective scattering. Three potentials used for Cooper pairs, fermionic-atom dimers, and semiconductor excitons are considered. While the s-wave scattering length for the BCS-like potential is just equal to its Born value, the other two are substantially smaller. For fermionic-atom dimers and semiconductor excitons, our results, calculated within a restricted correlated-pair basis, are in good agreement with those obtained from procedures numerically more demanding. We also propose model coboson-coboson scatterings that are separable and thus easily workable and that produce scattering lengths which match quantitatively well with the numerically obtained values for all fermion mass ratios. These separable model scatterings can facilitate future works on many-body effects in coboson gases.