To better constrain the physical mechanisms driving star formation, we present the first systematic study of the radio continuum size evolution of star-forming galaxies (SFGs) over the redshift range 0.35 < z < 2.25. We use the VLA COSMOS 3 GHz map (noise rms = 2.3 μJy beam⁻¹, θ_beam = 0.75 arcsec) to construct a mass-complete sample of 3184 radio-selected SFGs that reside on and above the main sequence (MS) of SFGs. We constrain the overall extent of star formation activity in galaxies by applying a 2D Gaussian model to their radio continuum emission. Extensive Monte Carlo simulations are used to validate the robustness of our measurements and characterize the selection function. We find no clear dependence between the radio size and stellar mass, M_⋆, of SFGs with 10.5 ≲ log(M_⋆/M_⊙) ≲ 11.5. Our analysis suggests that MS galaxies are preferentially extended, while SFGs above the MS are always compact. The median effective radius of SFGs on (above) the MS of R_eff = 1.5 ± 0.2 (1.0 ± 0.2) kpc remains nearly constant with cosmic time; a parametrization of the form R_eff ∝ (1 + z)^α yields a shallow slope of only α = −0.26 ± 0.08 (0.12 ± 0.14) for SFGs on (above) the MS. The size of the stellar component of galaxies is larger than the extent of the radio continuum emission by a factor ∼2 (1.3) at z = 0.5 (2), indicating star formation is enhanced at small radii. The galactic-averaged star formation rate surface density (Σ_SFR) scales with the distance to the MS, except for a fraction of MS galaxies (≲10%) that harbor starburst-like Σ_SFR. These “hidden” starbursts might have experienced a compaction phase due to disk instability and/or a merger-driven burst of star formation, which may or may not significantly offset a galaxy from the MS. We thus propose to use Σ_SFR and distance to the MS in conjunction to better identify the galaxy population undergoing a starbursting phase.