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.
Ian (Carl)’s Answer
The best way is to call or email your local clinic or hospital; most have shadowing programs that allow students to come in and observe there therapists.
G. Mark’s Answer
Whenever shadowing any professional, you would first have the subject asked if they would mind you observing the process. Assuming they agreed, you would essentially observe and not interfere, using the opportunity to take notes and do just that -- observe.
In my case, I was observing folks in their daily profession activity. The folks I observed were generally confident and experienced. I did not shadow anyone who had not had the experience of being observed before. And I tried to make myself as unobtrusive as possible.
This is a tremendous opportunity to gain insight to a professional in their, so to speak, "natural habitat". It is quite different from classwork and other training, and not to be missed.
Typically you would need to do it while in school. When I was getting my Bachelor's, I did an internship at a domestic violence center and was allowed to sit in on the group sessions. You may be able to do this as a volunteer, as well. Individual therapy tends to be more protected and you would likely need to be in grad school again doing an internship. As the others stated, clients need to be aware, sign releases, etc. Lots of ethical issues to take into consideration. That being said, there are also great videos out there of therapists doing individual, couples, and family therapy (this is how we started the training). Check out some of the more well known therapists or founders of the different types of therapy (Beck, Rogers, Gottman, etc). If you want to see a group-like process (that isn't typically led by a therapist, but can give you an idea of others sharing, encouraging, and even resisting treatment, go to an AA or NA meeting. They are free and have meetings that are "open" for people supporting the addict.