Is every training session checked?

Yes and no. Every training is checked.

But

The trainer checks them to a greater or lesser extent. 

To a lesser extent? It can only check the executed time.

If the athlete provides such data, the coach visually assesses the chart of each workout, whether the next sections of the workout are visible on it.

To a greater extent? If the trainer thinks that you may not have understood the training, he analyzes the watts/heart rate carefully and gives you tips on how to improve.