The lack of clean speech is a practical challenge to the development of speech enhancement systems, which means that the training of neural network models must be done in an unsupervised manner, and there is an inevitable mismatch between their training criterion and evaluation metric. In response to this unfavorable situation, we propose a teacher-student training strategy that does not require any subjective/objective speech quality metrics as learning reference by improving the previously proposed noisy-target training (NyTT). Because homogeneity between in-domain noise and extraneous noise is the key to the effectiveness of NyTT, we train various student models by remixing the teacher model's estimated speech and noise for clean-target training or raw noisy speech and the teacher model's estimated noise for noisy-target training. We use the NyTT model as the initial teacher model. Experimental results show that our proposed method outperforms several baselines, especially with two-stage inference, where clean speech is derived successively through the bootstrap model and the final student model.