Spectrum sharing offers a solution to the limited availability of spectrum in highly demanding wireless networks. However, sharing spectrum among multiple wireless nodes must not violate interference regulations in licensed networks, which is particularly difficult for uncoordinated networks. In this paper, we propose a novel approach that utilizes digital twins (DTs) of the physical environments to coordinate networks and approximate wireless channels, essential for maximizing transmission opportunities in an unlicensed network. To address the possible digital twin channel approximation errors, which may lead to more interference than planned on the licensed networks, we propose two robust beamforming designs for unlicensed users. Our results demonstrate that our conservative and probabilistic robust beamforming designs successfully meet interference constraints over 97.5% of the time across different levels of channel errors. Additionally, the conservative design can handle user localization errors at least 95% of the time. All that renders digital twins as a promising approach for spectrum sharing and coexistence.