Analog compute-in-memory (CIM) systems are promising for deep neural network (DNN) inference acceleration due to their energy efficiency and high throughput. However, as the use of DNNs expands, protecting user input privacy has become increasingly important. In this paper, we identify a potential security vulnerability wherein an adversary can reconstruct the user's private input data from a power side-channel attack, under proper data acquisition and pre-processing, even without knowledge of the DNN model. We further demonstrate a machine learning-based attack approach using a generative adversarial network (GAN) to enhance the data reconstruction. Our results show that the attack methodology is effective in reconstructing user inputs from analog CIM accelerator power leakage, even at large noise levels and after countermeasures are applied. Specifically, we demonstrate the efficacy of our approach on an example of U-Net inference chip for brain tumor detection, and show the original magnetic resonance imaging (MRI) medical images can be successfully reconstructed even at a noise-level of 20% standard deviation of the maximum power signal value. Our study highlights a potential security vulnerability in analog CIM accelerators and raises awareness of using GAN to breach user privacy in such systems.