How to decide "I'll trust this software" for closed-source or precompiled software?

Trust is not a boolean variable, "trusted = true / false", you should better think about trust level.

A few example of questions which may help you to evaluate the trust level you can grant to this software:

  1. How much do you trust the editor of this software?
  2. Could the software have been modified by a malicious 3rd-party between being created and being delivered to your computer?
  3. What is the sensitivity of the data you need provide to this software?
  4. What is the sensitivity of the data residing on the computer which will run this software?
  5. How long and how often will need to use this software?

If I correctly understand your question:

  1. You do not trust the editor, otherwise you wouldn't have asked this question in the first place,
  2. This software will just need the information related to this webinar you will attend,
  3. Your computer hosts sensitive or at least personal information which makes you worry about trust issues,
  4. This will be a one spot usage for this webinar, at best for further reference only.

In such conditions, I would just create some virtual machine so I would not worry anymore with any privacy issue while being free to comply with the webinar requests. Once the webinar ends, I will be free to either archive the VM image or drop it.