I have done it. I sign consent for my doctors to talk to each other when I feel comfortable.
It has been helpful when I started with a new T so she knew some backstory and I didn't have to start back at the why are you here, this is what therapy is discussion. If there is something you do not feel comfortable talking about or it is best to handle something a certain way, they can clue them in on that. For example, if there was prior abuse that you were not comfortable discussing, old T could just let T know that and make suggestions for how to best support you.
In my experience when I went from old T to new T she gave her some basic info regarding why I was there, how long, diagnoses, treatment and such. Then more information on what had been happening recently (like what the pressing issue was).
I used to be very wary and controlling but have realized that allowing for communication can (in the right circumstances) make things much easier. New T gets unbiased info, not clouded by emotions and I have to say less to new T (its hard for me to speak to new T's/people).
This was kinda long but I will end with this: Therapists are professionals. Most are able to be honest and frank with each other without fear of the reaction. They aren't invested to the level that we are.