I’ll reply without reading the other answers.
It feels quite similar to how I think about algorithms, in particular about brute-force algorithms, but it also has something similar to what I feel when trying to draw something. The first in that I try to think of a goal, then try to go through the steps to reach that goal and recurse when it does not work. The second in that I have to visualise the position on the board to judge the position: I can’t discover an atari without “seeing” the stones.
My eyes follow each move that is played, and my brain puts an imaginary stone on the place where it belongs. This is similar to what I do when trying to work out an algorithm or prove something mathematical: I see the object before me, and manipulate it in my mind towards some preferential goal.
However, the goal is not so clear with go as with algorithms, thus it feels more like drawing, where I try to visualise the finished drawing on top of the partial one, and judge if it will look alright.
Of course, as with both, I can only imagine things until a certain point, hence with drawing you sketch to help your imagination, and with algorithms / maths I almost always use a piece of paper to track progress with a difficult problem.
I also have to admit that thinking about a difficult math / programming problem kind of feels like I have to “start up” a section of my brain to dedicate to it. It’s pretty darn difficult to get this part of my brain “running”, but once it works, it’s quite difficult to get distracted. I have not felt that same kind of concentration when playing go, except for during byo-yomi and during the Thue-Morse Go matches. Oh, and it definitely starts up during games with equal / strong opponents in real life.
Now that I read the other comments, I also have a ‘voice’, and I ‘hear’ a sound for each stone, which helps remembering the sequence. Often it’s not the actual sound of stones, but more some kind of vague hum hum hum.
I never have dialogues in my head, only monologues, were I discuss things with myself. I do have different voices when thinking about things other people would say, but not for my own reasoning.