Personalized speech enhancement (PSE) models utilize additional cues, such as speaker embeddings like d-vectors, to remove background noise and interfering speech in real-time and thus improve the speech quality of online video conferencing systems for various acoustic scenarios. In this work, we propose two neural networks for PSE that achieve superior performance to the previously proposed VoiceFilter. In addition, we create test sets that capture a variety of scenarios that users can encounter during video conferencing. Furthermore, we propose a new metric to measure the target speaker over-suppression (TSOS) problem, which was not sufficiently investigated before despite its critical importance in deployment. Besides, we propose multi-task training with a speech recognition back-end. Our results show that the proposed models can yield better speech recognition accuracy, speech intelligibility, and perceptual quality than the baseline models, and the multi-task training can alleviate the TSOS issue in addition to improving the speech recognition accuracy.