We report the discovery and characterization of a pair of sub-Neptunes transiting the bright K-dwarf TOI-1064 (TIC 79748331), initially detected in the Transiting Exoplanet Survey Satellite (TESS) photometry. To characterize the system, we performed and retrieved the CHaracterising ExOPlanets Satellite (CHEOPS), TESS, and ground-based photometry, the High Accuracy Radial velocity Planet Searcher (HARPS) high-resolution spectroscopy, and Gemini speckle imaging. We characterize the host star and determine $T_{\rm eff, \star }=4734\pm 67\,\mathrm{ K}$, $R_{\star }=0.726\pm 0.007\, \mathrm{ R}_{\odot }$, and $M_{\star }=0.748\pm 0.032\, \mathrm{ M}_{\odot }$. We present a novel detrending method based on point spread function shape-change modelling and demonstrate its suitability to correct flux variations in CHEOPS data. We confirm the planetary nature of both bodies and find that TOI-1064 b has an orbital period of Pb = 6.44387 ± 0.00003 d, a radius of Rb = 2.59 ± 0.04 R⊕, and a mass of $M_{\rm b} = 13.5_{-1.8}^{+1.7}$ M⊕, whilst TOI-1064 c has an orbital period of $P_{\rm c} = 12.22657^{+0.00005}_{-0.00004}$ d, a radius of Rc = 2.65 ± 0.04 R⊕, and a 3σ upper mass limit of 8.5 M⊕. From the high-precision photometry we obtain radius uncertainties of ~1.6 per cent, allowing us to conduct internal structure and atmospheric escape modelling. TOI-1064 b is one of the densest, well-characterized sub-Neptunes, with a tenuous atmosphere that can be explained by the loss of a primordial envelope following migration through the protoplanetary disc. It is likely that TOI-1064 c has an extended atmosphere due to the tentative low density, however further radial velocities are needed to confirm this scenario and the similar radii, different masses nature of this system. The high-precision data and modelling of TOI-1064 b are important for planets in this region of mass-radius space, and it allow us to identify a trend in bulk density-stellar metallicity for massive sub-Neptunes that may hint at the formation of this population of planets.