Supernumerary robotic limbs (SRLs) are robotic structures integrated closely with the user's body, which augment human physical capabilities and necessitate seamless, naturalistic human-machine interaction. For effective assistance in physical tasks, enabling SRLs to hand over objects to humans is crucial. Yet, designing heuristic-based policies for robots is time-consuming, difficult to generalize across tasks, and results in less human-like motion. When trained with proper datasets, generative models are powerful alternatives for creating naturalistic handover motions. We introduce 3HANDS, a novel dataset of object handover interactions between a participant performing a daily activity and another participant enacting a hip-mounted SRL in a naturalistic manner. 3HANDS captures the unique characteristics of SRL interactions: operating in intimate personal space with asymmetric object origins, implicit motion synchronization, and the user's engagement in a primary task during the handover. To demonstrate the effectiveness of our dataset, we present three models: one that generates naturalistic handover trajectories, another that determines the appropriate handover endpoints, and a third that predicts the moment to initiate a handover. In a user study (N=10), we compare the handover interaction performed with our method compared to a baseline. The findings show that our method was perceived as significantly more natural, less physically demanding, and more comfortable.