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We examine the role of self-interaction errors (SIE) removal on the evaluation of magnetic exchange coupling constants. In particular we analyze the effect of scaling down the self-interaction-correction (SIC) for three {\em non-empirical} density functional approximations (DFAs) namely, the local spin density approximation, the Perdew-Burke-Ernzerhof generalized gradient approximation, and recent SCAN family of meta-GGA functionals. To this end, we employ three one-electron SIC methods: Perdew-Zunger [Perdew, J. P.; Zunger, A. \textit{Phys. Rev. B}, {\bf 1981}, \textit{23}, 5048] SIC, the orbitalwise scaled SIC method [Vydrov, O. A. \textit{et al.}, \textit{J. Chem. Phys.} {\bf 2006,} \textit{124}, 094108], and the recent {local} scaling method [Zope, R. R. \textit{et al.}, \textit{J. Chem. Phys.} {\bf 2019}, \textit{151}, 214108]. We compute the magnetic exchange coupling constants using the spin projection and non projection approaches for sets of molecules composed of dinuclear and polynuclear H--He models, organic radical molecules, and chlorocuprate, and compare these results against accurate theories and experiment. Our results show that for the systems that mainly consist of single electron regions, PZSIC performs well but for more complex organic systems and the chlorcuprates, an overcorrecting tendency of PZSIC combined with the DFAs utilized in this work is more pronounced, and in such cases LSIC with kinetic energy density ratio performs better than PZSIC. Analysis of the results in terms of SIC corrections to the density and to the total energy shows that both density and energy correction are required to obtain an improved prediction of magnetic exchange couplings.