We study the spin Coulomb drag in a quasi-two-dimensional electron gas of finite transverse width, including local field corrections beyond the random phase approximation (RPA). We find that the finite transverse width of the electron gas causes a significant reduction of the spin Coulomb drag. This reduction, however, is largely compensated by the enhancement coming from the inclusion of many-body local field effects beyond the RPA, thereby restoring good agreement with the experimental observations by C. P. Weber et al. [Nature (London) 437, 1330 (2005)].