Isolation Forest Prediction
The training dataset is only used to fit/train the model. The training will extract information from your training data and store it in your model so after training you no longer need it. You should never include training data when you later check the performance/accuracy of your model, that is only the test set. 

For predictions and computation of anomaly scores you only need to use the dataset you actually want to check for anomalies/outliers.

For streaming you should train first on offline data which only contains inliers or has a know ratio of outliers. Once it is trained you can use it to make predictions on new streamed data without comparing to your offline data.