Deep learning offers a promising solution to improve spectrum access techniques by utilizing data-driven approaches to manage and share limited spectrum resources for emerging applications. For several of these applications, the sensitive wireless data (such as spectrograms) are stored in a shared database or multistakeholder cloud environment and are therefore prone to privacy leaks. This paper aims to address such privacy concerns by examining the representative case study of shared database scenarios in 5G Open Radio Access Network (O-RAN) networks where we have a shared database within the near-real-time (near-RT) RAN intelligent controller. We focus on securing the data that can be used by machine learning (ML) models for spectrum sharing and interference mitigation applications without compromising the model and network performances. The underlying idea is to leverage a (i) Shuffling-based learnable encryption technique to encrypt the data, following which, (ii) employ a custom Vision transformer (ViT) as the trained ML model that is capable of performing accurate inferences on such encrypted data. The paper offers a thorough analysis and comparisons with analogous convolutional neural networks (CNN) as well as deeper architectures (such as ResNet-50) as baselines. Our experiments showcase that the proposed approach significantly outperforms the baseline CNN with an improvement of 24.5% and 23.9% for the percent accuracy and F1-Score respectively when operated on encrypted data. Though deeper ResNet-50 architecture is obtained as a slightly more accurate model, with an increase of 4.4%, the proposed approach boasts a reduction of parameters by 99.32%, and thus, offers a much-improved prediction time by nearly 60%.