What is your responsibility as an individual when viewing a webcast?
There are many ways in which to answer this question, because there are various situations.
If you are watching a webcast with a young person (minor), then you should know what the webcast is about. In other words, you would need to determine the suitability of the webcast. In this situation you are responsible for the minor who is watching the webcast with you.
If you are watching a webcast, then you are responsible for your decision for watching the webcast. The only exception that I can think of is if the webcast has inappropriate material that is not announced. Often shows offer a warning if something inappropriate is going to come on. For example, they say something like: "viewer discretion advised."