We present new CO(2-1) observations of a representative sample of 24 local (z < 0.02) luminous infrared galaxies (LIRGs) at high spa- tial resolution ( < 100 pc) from the Atacama Large Millimeter / submillimeter Array (ALMA). Our LIRGs lie above the main sequence (MS), with typical stellar masses in the range 10 10 -10 11 M (cid:12) and SFR ∼ 30 M (cid:12) yr − 1 . We derive the e ff ective radii of the CO(2-1) and the 1.3 mm continuum emissions using the curve-of-growth method. LIRGs show an extremely compact cold molecular gas distribution (median R CO ∼ 0.7 kpc), which is a factor 2 smaller than the ionized gas (median R H α ∼ 1.4 kpc), and 3.5 times smaller than the stellar size (median R star ∼ 2.4 kpc). The molecular size of LIRGs is similar to that of early-type galaxies (ETGs; R CO ∼ 1 kpc) and about a factor of 6 more compact than local spiral galaxies of similar stellar mass. Only the CO emission in low-z ULIRGs is more compact than these local LIRGs by a factor of 2. Compared to high-z (1 < z < 6) systems, the stellar sizes and masses of local LIRGs are similar to those of high-z MS star-forming galaxies (SFGs) and about a factor of 2-3 lower than submillimeter (submm) galaxies (SMGs). The molecular sizes of high-z MS SFGs and SMGs are larger than those derived for LIRGs by a factor of ∼ 3 and ∼ 8, respectively. Contrary to high-z SFGs and SMGs, which have comparable molecular and stellar sizes (median R star / R CO = 1.8 and 1.2, respectively), local LIRGs show more centrally concentrated molecular gas distribution (median R star / R CO = 3.3). fraction of the low-z LIRGs and high-z galaxies share a similar range in the size of the ionized gas distribution, from 1 to 4 kpc. However, no LIRGs with a very extended (above 4 kpc) radius are identified, while for high-z galaxies no compact (less than 1 kpc) emission is detected. These results indicate that while low-z LIRGs and high-z MS SFGs have similar stellar masses and sizes, the regions of current star formation (traced by the ionized gas) and of potential star formation (traced by the molecular gas) are substantially smaller in LIRGs, and constrained to the central kiloparsec (kpc) region. High-z galaxies represent a wider population but their star-forming regions are more extended, even covering the entire extent of the galaxy. High-z galaxies have larger fractions of gas than low-z LIRGs, and therefore the formation of stars could be induced by interactions and mergers in extended disks or filaments with su ffi ciently large molecular gas surface density involving physical mechanisms similar to those identified in the central kpc of LIRGs.