In this work, we examine the potential of autonomous operation of a reconfigurable intelligent surface (RIS) using wireless energy harvesting from information signals. To this end, we first identify the main RIS power-consuming components and introduce a suitable power-consumption model. Subsequently, we introduce a novel RIS power-splitting architecture that enables simultaneous energy harvesting and beamsteering. Specifically, a subset of the RIS unit cells (UCs) is used for beamsteering while the remaining ones absorb energy. For the subset allocation, we propose policies obtained as solutions to two optimization problems. The first problem aims at maximizing the signal-to-noise ratio (SNR) at the receiver without violating the RIS's energy harvesting demands. Additionally, the objective of the second problem is to maximize the RIS harvested power, while ensuring an acceptable SNR at the receiver. We prove that under particular propagation conditions, some of the proposed policies deliver the optimal solution of the two problems. Furthermore, we report numerical results that reveal the efficiency of the policies with respect to the optimal and very high-complexity brute-force design approach. Finally, through a case study of user tracking, we showcase that the RIS power-consumption demands can be secured by harvesting energy from information signals.