Visible crew/equipment: Cary Grant is crouching in his bunk bed in the train. The lights of the studio get reflected on his broken sunglasses when he shows them to Eve.
Visible crew/equipment: When Cary Grant and his film-mother enter the elevator at the Plaza this is filmed through a glass-door and you can see the reflection of a crewmember in a white shirt who is crouching below or next to the camera.
Visible crew/equipment: Right before Thornhill gets shot by Eve, he's talking to Vandamm at the table. The camera moves backward. You can see its shadow on Thornhill's arm.
Visible crew/equipment: When Thornhill exits the phone booth 6 stage lights are reflected on the glass door.
Visible crew/equipment: When the crop duster and the truck are about to explode and the truck drivers run away, one can tell it's not Cary Grant watching the havoc but his stunt double because of his rounder face and slightly different hairstyle. Also confirmed by actress Eve Saint-Marie.
Answer: Not really. You could (and at some hotels are still able to) keep your room number private or you could not - i.e. you could ask the hotel staff to keep your number secret from strangers, or you could ask them to tell anyone who might ask. Not having seen this movie, I don't know how likely it would be in the situations you speak of that the hotel guest would choose the latter option- it might be a mistake.
Blibbetyblip