This is a simple way to create what looks like overlapping circles. The problem is with circles, you have to send each circle to the back, and you never get the last overlap set that you want. So here is the trick!!
Start with a circle. Draw while holding SHIFT to create a perfect circle. Add a stroke to the shape, 10 pts work well.
Duplicate the circle, hold OPTION and click and drag, while holding SHIFT, and drag to the lower left. SHIFT will constrain to the left at a 45° angle.
Line up the duplicated circle to the point shown. Smart Guides will help in this alignment. View>Guides>Smart Guides
Use the pathfinder panel to Minus Front. This will remove the circle in front and leave the back circle shape.
OPTION + CLICK and drag the shape while holding SHIFT and drag at a 45° angle down and to the left. Snap the edges of the duplicated shape to the edges of the original shape.
Rotate the duplicated shape 90° clockwise.
Duplicate that shape and move up and to the left and snap to the second shape.
Rotate the newly duplicated shape 90° clockwise. Repeat the duplicate process up and to the right, hold shift and snap it to the existing circle. Rotate that shape 90° clockwise again.
With the last shape duplicated, this will drop right into final position if you have created perfect circles AND lines the circles up to the existing edges of the circles.
In Outline Mode (View>Outline) you should see all the shapes line up as shown.
Now you have an overlap composite shape.
Duplicate the top shape. Remove the stroke on it.
Choose Object>Offset Path and set the Offset to be a negative number. This will create a path inside the shape.
With the Offset Path, remove the OUTER path leaving the smaller, inner shape. Fill with a lighter color that you used in the original shape.
Move that smaller shape into position on the overlap shape.
Duplicate the inner shape, then rotate each 90° clockwise and move into position on the original shapes, aligning the center of the shapes.
If you change the stroke on the outer shapes to white, it changes the look considerably.
If you select everything and use the Pathfinder Minus Front Mode, this is the result.
If you select everything and use the Pathfinder Divide Mode, this is the result.