Under the organization of the base station (BS), wireless federated learning (FL) enables collaborative model training among multiple devices. However, the BS is merely responsible for aggregating local updates during the training process, which incurs a waste of the computational resource at the BS. To tackle this issue, we propose a semi-federated learning (SemiFL) paradigm to leverage the computing capabilities of both the BS and devices for a hybrid implementation of centralized learning (CL) and FL. Specifically, each device sends both local gradients and data samples to the BS for training a shared global model. To improve communication efficiency over the same time-frequency resources, we integrate over-the-air computation for aggregation and non-orthogonal multiple access for transmission by designing a novel transceiver structure. To gain deep insights, we conduct convergence analysis by deriving a closed-form optimality gap for SemiFL and extend the result to two extra cases. In the first case, the BS uses all accumulated data samples to calculate the CL gradient, while a decreasing learning rate is adopted in the second case. Our analytical results capture the destructive effect of wireless communication and show that both FL and CL are special cases of SemiFL. Then, we formulate a non-convex problem to reduce the optimality gap by jointly optimizing the transmit power and receive beamformers. Accordingly, we propose a two-stage algorithm to solve this intractable problem, in which we provide the closed-form solutions to the beamformers. Extensive simulation results on two real-world datasets corroborate our theoretical analysis, and show that the proposed SemiFL outperforms conventional FL and achieves 3.2% accuracy gain on the MNIST dataset compared to state-of-the-art benchmarks.