Context. The earliest evolutionary stages of low-mass protostars are characterised by hot and fast jets which remove angular momentum from the circumstellar disk, thus allowing mass accretion onto the central object. However, the launch mechanism is still being debated. Aims. We would like to exploit high-angular (∼0. 8) resolution and high-sensitivity images to investigate the origin of protostellar jets using typical molecular tracers of shocked regions, such as SiO and SO. Methods. We mapped the inner 22 of the NGC 1333-IRAS2A protostar in SiO(5-4), SO(6 5 -5 4 ), and the continuum emission at 1.4 mm using the IRAM Plateau de Bure Interferometer in the framework of the CALYPSO IRAM large program. Results. For the first time, we disentangle the NGC 1333-IRAS2A Class 0 object into a proto-binary system revealing two protostars (MM1, MM2) separated by ∼560 AU, each of them driving their own jet, while past work considered a single protostar with a quadrupolar outflow. We reveal (i) a clumpy, fast (up to |V − V LSR | ≥ 50 km s −1 ), and blueshifted jet emerging from the brightest MM1 source; and (ii) a slower redshifted jet, driven by MM2. Silicon monoxide emission is a powerful tracer of high-excitation (T kin ≥ 100 K; n H 2 ≥ 10 5 cm −3 ) jets close to the launching region. At the highest velocities, SO appears to mimic SiO tracing the jets, whereas at velocities close to the systemic one, SO is dominated by extended emission, tracing the cavity opened by the jet. Conclusions. Both jets are intrinsically monopolar, and intermittent in time. The dynamical time of the SiO clumps is ≤30-90 yr, indicating that one-sided ejections from protostars can take place on these timescales.