Linda Ann’s Answer
I just finished reading the two other answers to this question. Neither of these men are in the psychology profession and therefore wouldn't be aware of the Ethical Code that governs how therapists go about their business.
It is not likely that you would be able to shadow a therapist until you were enrolled in at least a master's degree counseling or psychology program AND the patient consented to your presence in the therapy room OR via one-way mirror OR via a video of a session.
You would not be permitted to take notes, again unless the client (patient) gave express, written permission. What occurs in a therapeutic session is "sacred." What is said stays in the room; any notes or recordings that the therapist takes must remain under lock and key, per our Ethical Code.
So, you need to be enrolled in a graduate program, generally, in order to engage in such shadowing.