This paper presents a private-partial distributed least mean square (PP-DLMS) algorithm that offers energy efficiency while preserving privacy and is suitable for applications with limited resources and strict security requirements. The proposed PP-DLMS allows every agent to exchange only a fraction of their perturbed data with neighbors during the collaboration process to minimize communication costs and guarantee privacy simultaneously. In order to understand how partial-sharing of perturbed data affects the learning performance, we conduct mean convergence analysis. Moreover, to investigate the privacy-preserving properties of the proposed algorithm, we characterize agent privacy in the presence of an honest-but-curious (HBC) adversary. Analytical results show that the proposed PP-DLMS is resilient against an HBC adversary by providing a fair energy-privacy trade-off compared to the conventional LMS algorithm. Numerical simulations corroborate the analytical findings.