This is indeed a complex question, but here's a brief set of answers:
Are there any risks in using these online chat channels?
Whenever anything is implemented over the Web, there are always risks associated. One of the risks you will encounter with chats sessions for uses as described is information confidentiality.
If so, what are the primary risks?
Information confidentiality. I have had cases in the past when, while chatting with an agent to resolve an issue I had, he/she accidentally posted private information related to a different customer.
Can the chat website view any of our private information, and if so, what?
This will depend on how the chat service is implemented. There are applications out there that would take care of some of this problems for you, if you decide to use them.
Also, is it secure to share private and confidential information on those chat channels?
Simply put, no. Again, information control would play a role on what the support analyst is capable of viewing, and disclosing.