How to combine two logistic regression models trained on different set of data?
I can think of two options, there are probably more:

Train a binary classification &quot;top model&quot; which predicts whether the instance belongs to  model 1 or model 2. The two final models only need to be trained with their respective data and only predict their respective N or M classes. The performance of the top model is crucial in this option.
Train an open set classification model. It's a bit similar to what you're doing currently by adding a class &quot;other model&quot; to the first model, except that &quot;other&quot; is not a regular class and you don't need to train the model with instances from this class. This is related to one-class classification and requires a specific training method which can handle open-set classification.