The 0/1 values are nominal in logistic regression, hence they are treated as two cases rather than numbers (you may use any label e.g. apples/oranges and get the same curve). In JMP the data in the logistic plot (Fit Y by X platform) seem to be randomly distributed within their area, i.e. all 0 below and all 1 above the sigmoid curve. The right Y-axis is not metric scale from 0-1 but indicates the total proportions of the two cases.
However, it may be possible to get the type of plot you want by using the Nonlinear platform, and defining the logistic equation manually or from the model library.
Use nonlinear platform. The table need three columns: X (independent cont. variable), Y (0/1 or probabilities between 0 and 1) , and a formula column predicting Y from X using e.g. your logistic equation.