Personalized news recommendation has been widely adopted to improve user experience. Recently, pre-trained language models (PLMs) have demonstrated the great capability of natural language understanding and the potential of improving news modeling for news recommendation. However, existing PLMs are usually pre-trained on general corpus such as BookCorpus and Wikipedia, which have some gaps with the news domain. Directly finetuning PLMs with the news recommendation task may be sub-optimal for news understanding. Besides, PLMs usually contain a large volume of parameters and have high computational overhead, which imposes a great burden on the low-latency online services. In this paper, we propose Tiny-NewsRec, which can improve both the effectiveness and the efficiency of PLM-based news recommendation. In order to reduce the domain gap between general corpora and the news data, we propose a self-supervised domain-specific post-training method to adapt the generally pre-trained language models to the news domain with the task of news title and news body matching. To improve the efficiency of PLM-based news recommendation while maintaining the performance, we propose a two-stage knowledge distillation method. In the first stage, we use the domain-specific teacher PLM to guide the student model for news semantic modeling. In the second stage, we use a multi-teacher knowledge distillation framework to transfer the comprehensive knowledge from a set of teacher models finetuned for news recommendation to the student. Experiments on two real-world datasets show that our methods can achieve better performance in news recommendation with smaller models.