Do SSI see if I open another bank account?

They might - in terms of technology, they can arrange some form of automatic communication with banks which raises flags whenever something happens to an individual of interest (ie someone receiving stipend).
Whether they're allowed to by laws of privacy, or they actually have done so, I cannot tell. I'd suspect they have you sign a waiver as part of the processfor the former, and yes for the latter given perceived risk of fraud.