Federated learning (FL) naturally faces the problem of data heterogeneity in real-world scenarios, but this is often overlooked by studies on FL security and privacy. On the one hand, the effectiveness of backdoor attacks on FL may drop significantly under non-IID scenarios. On the other hand, malicious clients may steal private data through privacy inference attacks. Therefore, it is necessary to have a comprehensive perspective of data heterogeneity, backdoor, and privacy inference. In this paper, we propose a novel privacy inference-empowered stealthy backdoor attack (PI-SBA) scheme for FL under non-IID scenarios. Firstly, a diverse data reconstruction mechanism based on generative adversarial networks (GANs) is proposed to produce a supplementary dataset, which can improve the attacker's local data distribution and support more sophisticated strategies for backdoor attacks. Based on this, we design a source-specified backdoor learning (SSBL) strategy as a demonstration, allowing the adversary to arbitrarily specify which classes are susceptible to the backdoor trigger. Since the PI-SBA has an independent poisoned data synthesis process, it can be integrated into existing backdoor attacks to improve their effectiveness and stealthiness in non-IID scenarios. Extensive experiments based on MNIST, CIFAR10 and Youtube Aligned Face datasets demonstrate that the proposed PI-SBA scheme is effective in non-IID FL and stealthy against state-of-the-art defense methods.