They never actually contacted me directly to verify my right to request the notes... This was what happened for me for my requests:
1. I sent in the request through the ATIP link, and uploaded the consent form (I was requesting for my husband who is living in the USA)
2. I received an automatic e-mail confirmation after my payment was processed that my request was received
3. About 30 calendar days later, I received an e-mail containing the requested notes (CSIS is faster than CBSA; I think CSIS gets less requests)
I am a Canadian citizen and I actually never submitted proof of my citizenship in any of my ATIP requests for any of my sets of notes (IRCC, CBSA, and CSIS), even though it is listed as a required document (I forgot; it was not intentional). However, they still processed my requests without asking for this proof. However, I'm not sure if they would do the same for you if your qualifying status giving you right to request ATIP is different than mine.
Also, the first time that I made requests for CSIS and CBSA notes, I used the wrong consent forms - I used the IRCC consent form for both of them. After 1-2 weeks, I got an e-mail from a representative from each agency with a link to the correct consent form to use for them and telling me to get it filled out and e-mailed back to them. They seem like they are quite reasonable about asking for more information/proof/documents if they need it instead of just denying your request outright.
By the way, CBSA specifically notified me of some special COVID circumstances where the notes they give you in response to your request may not be complete - see this post
https://www.canadavisa.com/canada-immigration-discussion-board/threads/confusion-about-csis-notes.698398/post-8815047
Hope that helps!