Reconfigurable intelligent surfaces (RISs) are envisioned as a potentially transformative technology for future wireless communications. However, RISs' inability to process signals and the attendant increased channel dimension have brought new challenges to RIS-assisted systems, including significantly increased pilot overhead required for channel estimation. To address these problems, several prior contributions that enhance the hardware architecture of RISs or develop algorithms to exploit the channels' mathematical properties have been made, where the required pilot overhead is reduced to be proportional to the number of RIS elements. In this paper, we propose a dimension-independent channel state information (CSI) acquisition approach in which the required pilot overhead is independent of the number of RIS elements. Specifically, in contrast to traditional signal transmission methods, where signals from the base station (BS) and the users are transmitted in different time slots, we propose a novel method in which signals are transmitted from the BS and the user simultaneously during CSI acquisition. With this method, an electromagnetic interference random field (IRF) will be induced on the RIS, and we propose the structure of sensing RIS to capture its features. Moreover, we develop three algorithms for parameter estimation in this system, in which one of the proposed vM-EM algorithm is analyzed with the fixed-point perturbation method to obtain an asymptotic achievable bound. In addition, we also derive the Cramér-Rao lower bound (CRLB) and an asymptotic expression for characterizing the best possible performance of the proposed algorithms. Simulation results verify that our proposed signal transmission method and the corresponding algorithms can achieve dimension-independent CSI acquisition for beamforming.