So much antipathy towards the poor pigeons! I don't think the pigeons are stupid at all; I think they know that the peregrines normally go after them only when they are in flight. I also think that they can see whether a peregrine has a full crop, and realize that they are safe for a while. Unlike humans, peregrines don't hunt for so-called "sport".
Also, pigeons are not the only birds to approach peregrines. I remember seeing a little mockingbird go dancing along a branch towards Newton, the female at the I-90 Bridge site in Ohio. The little bird went right up to her, and started singing. When he finished his song, he danced back along the branch and left. Newton just looked at him.