I've been asked number of dives I've done, and voluntarily e-mailed my Excel log to my IDC Course Director (to become an instructor) and to the Instructor I did Divemaster with.
Never have I ever has somebody sign a log. The original YMCA logbook I got in 1987 didn't, as I recall, have a spot for somebody to sign.
More recently, I self-certified to be a Dry Suit Instructor (and a few others). For this, I provided PADI with snippets from the Excel sheet showing logged dives doing the various specialties. They never raised an eyebrow.
Signatures, though? Nope.
Edit: I wonder if you get looked at more closely if you barely hit the minimum. E.g., if you need X dives to take the Seaweed Rescue and Rehabilitation and you only have X+1 dives you claim to have logged you'll get looked at more closely than if you have 10X dives logged.