We present the eddington bias corrected specific star formation rate function (sSFRF) at different stellar mass scales from a subsample of the Sloan Digital Sky Survey Data Release DR7 (SDSS), which is considered complete both in terms of stellar mass (M-*) and star formation rate (SFR). The above enable us to study qualitatively and quantitatively quenching, the distribution of passive/star-forming galaxies and perform comparisons with the predictions from state-of-the-art cosmological models, within the same M-* and SFR limits. We find that at the low-mass end (M-* = 10(9.5)-10(10)M(circle dot)) the sSFRF is mostly dominated by star-forming objects. However, moving to the two more massive bins (M-* = 10(10)-10(10.5) M-circle dot and M* = 10(10.5)-10(11)M(circle dot)) a bimodality with two peaks emerges. One peak represents the star-forming population, while the other describes a rising passive population. The bi-modal form of the sSFRFs is not reproduced by a range of cosmological simulations (e.g. Illustris, EAGLE, Mufasa, and IllustrisTNG) which instead generate mostly the star-forming population, while a bi-modality emerges in others (e.g. L-Galaxies, Shark, and Simba). Our findings reflect the need for the employed quenching schemes in state-of-the-artmodels to be reconsidered, involving prescriptions that allow 'quenched galaxies' to retain a small level of SF activity (sSFR = 10(-11) 10(-12) yr(-1)) and generate an adequate passive population/bi-modality even at intermediate masses (M-* = 10(10)-10(10.5)M(circle dot)).