How can create effect of inner depth like on the picture?
The shadow is a blurred black ring. See the cartoon:
Have a grey background, a colored plate with a hole and 2 rings (White and black) which fit into the hole or preferably are a little larger.(see NOTE2) Holes can be made with Subtract in the pathfinder panel.
Select the black ring and goto Effects > Blur > Gaussian Blur (see NOTE1)
Remove the strokes, align the shapes and arrange them to right order, move manually the blurred ring a little aside.
If you want your shadow to have a sharper edge as in your example, use less blur and make the black ring partially transparent (= reduce Opacity in the Appearance panel). You can even have 2 rings with different blurs, placements and opacities. This is the way to make an exact copy of your example.
NOTE1: You can edit the blur afterwards. Select the blurred ring and goto the Appearance panel. Blur is there as a line. Click it!
NOTE2: In format conversions exactly fitting seams can lose their watertightness. It's no problem if you stay in Ai.
ADDENDUM due the comment:
Unfortunately I do not know the details of the conversion from ai to eps. You can fill the black ring with a gradient mesh or a radial gradient. Both allow transparency. You can for example have radial gradient from black to partially or full transparent black. Gradient from black to the background color does not need transparency at all. Check these.