We present the discovery and characterization of two warm mini-Neptunes transiting the K3V star TOI-815 in a K-M binary system. Analysis of the spectra and rotation period reveal it to be a young star with an age of 200^+400^_-200_Myr. TOI-815b has a 11.2-day period and a radius of 2.94+/-0.05 R_{Earth}_ with transits observed by TESS, CHEOPS, ASTEP, and LCOGT. The outer planet, TOI-815c, has a radius of 2.62+/-0.10R_{Earth}_, based on observations of three non-consecutive transits with TESS, while targeted CHEOPS photometry and radial velocity follow-up with ESPRESSO were required to confirm the 35-day period. ESPRESSO confirmed the planetary nature of both planets and measured masses of 7.6+/-1.5M_{Earth}_ ({rho}P=1.64^+0.33^_-0.31_g/cm^3^) and 23.5+/-2.4M_{Earth}_ ({rho}P=7.2^+1.1^_-1.0_g/cm^3^) respectively. Thus, the planets have very different masses, unlike the usual similarity of masses in compact multi-planet systems. Moreover, our statistical analysis of mini-Neptunes orbiting FGK stars suggests that weakly irradiated planets tend to have higher bulk densities compared to those suffering strong irradiation. This could be ascribed to their cooler atmospheres, which are more compressed and denser. Internal structure modeling of TOI-815b suggests it likely has a H-He atmosphere constituting a few percent of the total planet mass, or higher if the planet is assumed to have no water. In contrast, the measured mass and radius of TOI-815c can be explained without invoking any atmosphere, challenging planetary formation theories.