If you can manage the text part of logo to be expanded, you can try this approach:
Not sure of exact effect but one way of achieving similar effect is using a duplicate layer of the text as a Shadow Layer.
I don't know how this bent text is done in Illustrator, I used Object > Envelope Distort feature to achieve this effect, and then expanded the shape. Let's call it text layer.
Once you've the text layer. Duplicate it and Edit > Paste in Back and change the color to desired shadow color. You'll need it as a shadow layer.
Select the shadow layer and increase the size a bit (the amount you need for the shadow) and shift it a little down
- Apply a stroke to text layer and expand it again. I used a purple stroke (color wouldn't matter. See next step.)
Select and expand the text layer again.
Group everything (shadow layer + text layer) and use Shape Builder tool and carefully remove all unwanted shapes so only shadow and text remains.
- You should get a result like this: