To acquire high-quality positron emission tomography (PET) images while reducing the radiation tracer dose, numerous efforts have been devoted to reconstructing standard-dose PET (SPET) images from low-dose PET (LPET). However, the success of current fully-supervised approaches relies on abundant paired LPET and SPET images, which are often unavailable in clinic. Moreover, these methods often mix the dose-invariant content with dose level-related dose-specific details during reconstruction, resulting in distorted images. To alleviate these problems, in this paper, we propose a two-stage Semi-Supervised SPET reconstruction framework, namely S3PET, to accommodate the training of abundant unpaired and limited paired SPET and LPET images. Our S3PET involves an un-supervised pre-training stage (Stage I) to extract representations from unpaired images, and a supervised dose-aware reconstruction stage (Stage II) to achieve LPET-to-SPET reconstruction by transferring the dose-specific knowledge between paired images. Specifically, in stage I, two independent dose-specific masked autoencoders (DsMAEs) are adopted to comprehensively understand the unpaired SPET and LPET images. Then, in Stage II, the pre-trained DsMAEs are further finetuned using paired images. To prevent distortions in both content and details, we introduce two elaborate modules, i.e., a dose knowledge decouple module to disentangle the respective dose-specific and dose-invariant knowledge of LPET and SPET, and a dose-specific knowledge learning module to transfer the dose-specific information from SPET to LPET, thereby achieving high-quality SPET reconstruction from LPET images. Experiments on two datasets demonstrate that our S3PET achieves state-of-the-art performance quantitatively and qualitatively.