How to protect user privacy when conducting research on opening bank accounts?

I work for a bank and we want to redesign our online account opening process. We don't track analytics and we have limited insight into the user experience. We don't have the ability for people to open fake accounts for testing purposes, so we'll have to observe people as they open real accounts.

The user needs to input personal info (SSN, credit card numbers, payments, etc), so we need to respect their privacy. How do we respect user privacy while gathering good data? How do we put people at ease and encourage them to participate? Should we instead consider alternatives to direct user observations?

The plan is to observe users while sitting over their shoulder. I want to record the screen too but I don't want to make people uncomfortable. But that data would be very valuable so I'm not sure. My thought was to go to a bank branch and try to recruit customers when they mention to tellers that they'd like to open an account. Customers often go to the same tellers every time so they trust them. I'm hoping that the tellers can be our liaisons and put people at ease when they ask them to volunteer for research.

Thanks in advance!