The origin and evolution of the sub-Neptune population is a highly debated topic in the exoplanet community. With the advent of JWST, atmospheric studies can now put unprecedented constraints on the internal composition of this population. In this context, the THIRSTEE project aims to investigate the population properties of sub-Neptunes with a comprehensive and demographic approach, providing a homogeneous sample of precisely characterised sub-Neptunes across stellar spectral types. We present here the precise characterisation of the planetary system orbiting one of the THIRSTEE M-dwarf targets, TOI-771 (d=25pc, V=14.9mag), known to host one planet, TOI-771 b, which has been statistically validated using TESS observations. We use TESS, SPECULOOS, TRAPPIST and M-Earth photometry together with 31 high-precision ESPRESSO radial velocities to derive the orbital parameters and investigate the internal composition of TOI-771 b, as well as exploring the presence of additional companions in the system. We derive precise mass and radius for TOI-771 b, a super-Earth with R_b_=1.36+/-0.10R_{Earth}_ and M_b_=2.47+/-0.32M_{Earth}_ orbiting every 2.3 days around its host star. Its composition is consistent with an Earth-like planet, and it adds up to the rocky population of sub-Neptunes lying below the density gap identified around M dwarfs. With a 13% precision in mass, a 7% radius precision, and a warm equilibrium temperature of Teq=543K, TOI-771 b is a particularly interesting target for atmospheric characterisation with JWST, and it is indeed one of the targets under consideration for the Rocky World DDT program. Additionally, we discover the presence of a second, non-transiting planet in the system, TOI-771 c, with a period of 7.61 days and a minimum mass of Mpsin(i)=2.87^+0.41^_-0.38_M_{Earth}_. Even though the inclination is not directly constrained, the planet likely belongs to the temperate sub-Neptune population, with an equilibrium temperature of 365K.