We suggest an approach to automatic picking of P- and S-wave arrival times when processing data from local seismological-monitoring network. A distinctive feature of this approach is that it does not attempt to train a universal neural network for processing all types of seismological data. Instead, we focus on one specific region at a time, which significantly narrows the requirements for the training dataset size and variability. An important result is the automatic quality-control tool, since it simultaneously ensures the accuracy of the accepted events as well as forms a fairly small dataset of rejected events. This small dataset can be further used for manual processing and additional neural-network training. This approach was tested on real data from two local seismological networks located in different regions. We demonstrate that a small seismological dataset can be used for training the neural network for processing seismological data from a specific region: records from 2040 local earthquakes. For high-quality data, it is possible to pick the arrival times of P- and S-waves with an error less than 50 ms for 94 and 88% of cases, respectively. For the poor-quality dataset, it was possible to determine the arrival times of P- and S-waves with an error less than 200 ms in 82 and 73% of cases, respectively. The proposed approach makes it possible to accelerate automatic processing by reducing the required size of the training sample; the approach was implemented in stream processing for the considered seismological networks.