Improving our understanding of the nuclear properties of high-Eddington-ratio (λ_Edd) active galactic nuclei (AGN) is necessary since at this regime the radiation pressure is expected to affect the structure and efficiency of the accretion disc-corona system. This may cause departures from the typical nuclear properties of low-λ_Edd AGN, which have been largely studied so far. We present here the X-ray spectral analysis of 14 radio-quiet, λ_Edd ≳ 1 AGN at 0.4 ≤ z ≤ 0.75, observed with XMM-Newton. Optical/UV data from simultaneous Optical Monitor observations have also been considered. These quasars were selected to have relatively high values of black hole mass (M_BH ∼ 10^{8 − 8.5} M_⊙) and bolometric luminosity (L_bol ∼ 10⁴⁶ erg s⁻¹) in order to complement previous studies of high-λ_Edd AGN at lower M_BH and L_bol. We studied the relation between λ_Edd and other key X-ray spectral parameters, such as the photon index (Γ) of the power-law continuum, the X-ray bolometric correction (k_{bol, X}), and the optical/UV-to-X-ray spectral index (α_ox). Our analysis reveals that, despite the homogeneous optical and supermassive black hole accretion properties, the X-ray properties of these high-λ_Edd AGN are quite heterogeneous. We indeed measured values of Γ between 1.3 and 2.5, at odds with the expectations based on previously reported Γ − λ_Edd relations, for which Γ ≥ 2 would be a ubiquitous hallmark of AGN with λ_Edd ∼ 1. Interestingly, we found that ∼30% of the sources are X-ray weak, with an X-ray emission about a factor of ∼10 − 80 fainter than that of typical AGN at similar UV luminosities. The X-ray weakness seems to be intrinsic and not due to the presence of absorption along the line of sight to the nucleus. This result may indicate that high-λ_Edd AGN commonly undergo periods of intrinsic X-ray weakness. Furthermore, results from follow-up monitoring with Swift of one of these X-ray weak sources suggest that these periods can last for several years.