Context. Colliding collisionless shocks appear across a broad variety of astrophysical phenomena and are thought to be possible sources of particle acceleration in the Universe. Aims. The main goal of our experimental and computational work is to understand the effect of the interpenetration between two subcritical collisionless shocks on particle energization. Methods. To investigate the detailed dynamics of this phenomenon, we performed a dedicated laboratory experiment. We generated two counter-streaming subcritical collisionless magnetized shocks by irradiating two Teflon (C2F4) targets with 100 J, 1 ns laser beams on the LULI2000 laser facility. The interaction region between the plasma flows was pre-filled with a low-density background hydrogen plasma and initialized with an externally applied homogeneous magnetic field perpendicular to the shocks. We also modeled the macroscopic evolution of the system via hydrodynamic simulations and the microphysics at play during the interaction via particle-in-cell (PIC) simulations. Results. Here, we report our measurements of the plasma density and temperature during the formation of the supercritical shocks, their transition to subcritical, and their final interpenetration. We find that in the presence of two shocks, the ambient ions reach energies around 1.5 times of those obtained with single shocks. Both the presence of the downstream zone of the second shock and of the downstream zone common for the two shocks play a role in the different energization: the characteristics of the perpendicular electric fields in the two areas indeed allow for certain particles to continue being accelerated or, at least, to avoid being decelerated. Conclusions. The findings of our laboratory investigation are relevant for our understanding of the energy distribution of high-energy particles that populate the interplanetary space in our solar system and the very local interstellar medium around the heliopause, where observations have indicated evidence of subcritical collisionless shocks that may eventually go on to collide with one another.