This is relatively easy using a distance transform:

img = Import["https://i.sstatic.net/Juolc.png"]
binary = Binarize[img, .9];
dist = DistanceTransform[ColorNegate[binary]];

Now, dist, contains for each pixel the euclidean distance to the closest white pixel. The points you're looking for are (I think) the local maxima in this image:

maxPos = ComponentMeasurements[MaxDetect[dist], "Centroid"][[All, 2]];    
radii = PixelValue[dist, maxPos]

Which yields these circles:

Show[binary, Graphics[{Red, MapThread[Circle, {maxPos, radii}]}]]

and the radii:

Histogram[radii, {1}]

These are the circles that are locally maximal. So they're usually touching 3 or more white points. If you want all circles that touch two white points, you should look at SkeletonTransform.

For example, for a square, the method above wold yield only on circle, at the center of the square. SkeletonTransform would yield all points along the diagonals of the square - as circles centered on the diagonals touch the square in two places.