We propose an ensemble learning framework with Poisson sub-sampling to effectively train a collection of teacher models to issue some differential privacy (DP) guarantee for training data. Through boosting under DP, a student model derived from the training data suffers little model degradation from the models trained with no privacy protection. Our proposed solution leverages upon two mechanisms, namely: (i) a privacy budget amplification via Poisson sub-sampling to train a target prediction model that requires less noise to achieve a same level of privacy budget, and (ii) a combination of the sub-sampling technique and an ensemble teacher-student learning framework that introduces DP-preserving noise at the output of the teacher models and transfers DP-preserving properties via noisy labels. Privacy-preserving student models are then trained with the noisy labels to learn the knowledge with DP-protection from the teacher model ensemble. Experimental evidences on spoken command recognition and continuous speech recognition of Mandarin speech show that our proposed framework greatly outperforms existing DP-preserving algorithms in both speech processing tasks.