Context. Owing to the paucity of sub-arcsecond (sub)mm observations required to probe the innermost regions of newly forming protostars, several fundamental questions are still being debated, such as the existence and coevality of close multiple systems. Aims. We study the physical and chemical properties of the jets and protostellar sources in the NGC 1333-IRAS4A proto-binary system using continuum emission and molecular tracers of shocked gas. Methods. We observed NGC 1333-IRAS4A in the SiO(6−5), SO(6 5 −5 4 ), and CO(2−1) lines and the continuum emission at 1.3, 1.4, and 3 mm using the IRAM Plateau de Bure Interferometer in the framework of the CALYPSO large program. Results. We clearly disentangle for the first time the outflow emission from the two sources A1 and A2. The two protostellar jets have very different properties: the A1 jet is faster, has a short dynamical timescale ( 10 3 yr), and is associated with H 2 shocked emission, whereas the A2 jet, which dominates the large-scale emission, is associated with diffuse emission, bends, and emits at slower velocities. The observed bending of the A2 jet is consistent with the change of propagation direction observed at large scale and suggests jet precession on very short timescales (∼200−600 yr). In addition, a chemically rich spectrum with emission from several complex organic molecules (e.g. HCOOH, CH 3 OCHO, CH 3 OCH 3 ) is only detected towards A2. Finally, very high-velocity shocked emission (∼50 km s −1 ) is observed along the A1 jet. An LTE analysis shows that SiO, SO, and H 2 CO abundances in the gas phase are enhanced up to (3−4) × 10 −7 , (1.4−1.7) × 10 −6 , and (3−7.9) × 10 −7 , respectively. Conclusions. The intrinsic different properties of the jets and driving sources in NGC 1333-IRAS4A suggest different evolutionary stages for the two protostars, with A1 being younger than A2, in a very early stage of star formation previous to the hot-corino phase.