This paper introduces a practical approach for leveraging a real-time deep learning model to alternate between speech enhancement and joint speech enhancement and separation depending on whether the input mixture contains one or two active speakers. Scale-invariant signal-to-distortion ratio (SI-SDR) has shown to be a highly effective training measure in time-domain speech separation. However, the SI-SDR metric is ill-defined for zero-energy target signals, which is a problem when training a speech separation model using utterances with varying numbers of talkers. Unlike existing solutions that focus on modifying the loss function to accommodate zero-energy target signals, the proposed approach circumvents this problem by training the model to extract speech on both its output channels regardless if the input is a single or dual-talker mixture. A lightweight speaker overlap detection (SOD) module is also introduced to differentiate between single and dual-talker segments in real-time. The proposed module takes advantage of the new formulation by operating directly on the separated masks, given by the separation model, instead of the original mixture, thus effectively simplifying the detection task. Experimental results show that the proposed training approach outperforms existing solutions, and the SOD module exhibits high accuracy.