Renewable energy communities (RECs) are widely regarded as a transformative opportunity to enhance the management of electricity distribution networks, benefiting the system as a whole and its participants through local energy production, increased self-consumption, and empowering citizens. However, their proliferation introduces significant challenges for distribution system management, particularly at the low-voltage (LV) level, where participants are primarily located. Despite its critical role, the LV network is often overlooked in favor of studies focusing on the system-level impacts. This paper addresses this gap by evaluating the impact of RECs on LV networks and the broader distribution system. The study analyzes various LV networks representative of the Italian context, encompassing both rural and urban areas. By leveraging the software tool OpenDSS and Monte Carlo simulations over an entire year, the analysis captures the inherent variability of load demand and photovoltaic generation, as well as the resulting network imbalances under diverse policy scenarios. The findings reveal that the increasing level of self-consumption could significantly challenge distribution network operation, limiting also the sourcing of flexibility. These results underscore the necessity for advanced management strategies and targeted investments in grid flexibility to ensure the reliability and efficiency of distribution networks integrating RECs.