The privacy of data is a major challenge in machine learning as a trained model may expose sensitive information of the enclosed dataset. Besides, the limited computation capability and capacity of edge devices have made cloud-hosted inference inevitable. Sending private information to remote servers makes the privacy of inference also vulnerable because of susceptible communication channels or even untrustworthy hosts. In this paper, we target privacy-preserving training and inference of brain-inspired Hyperdimensional (HD) computing, a new learning algorithm that is gaining traction due to its light-weight computation and robustness particularly appealing for edge devices with tight constraints. Indeed, despite its promising attributes, HD computing has virtually no privacy due to its reversible computation. We present an accuracy-privacy trade-off method through meticulous quantization and pruning of hypervectors, the building blocks of HD, to realize a differentially private model as well as to obfuscate the information sent for cloud-hosted inference. Finally, we show how the proposed techniques can be also leveraged for efficient hardware implementation.