1) "Where(i) [ do you think [ (that) he went ____(i) afterwards] ] ?"
Let's first look at a somewhat related declarative clause version for that interrogative clause:
- 2) "You think [ (that) he went to the store afterwards]."
Now let's convert that to an interrogative clause with the interrogative word in situ:
- 3) "You think [ (that) he went where afterwards] ?"
Now let's front that interrogative word:
- 4) "Where(i) do [you think [ (that) he went ____(i) afterwards] ] ?"
Basically, versions #3 and #4 will usually be interpreted to have the same meaning.
As to what goes through the reader's (or listener's) head when reading (or hearing) your original example, I'd suppose that it would be similar to whatever happens when processing fronted elements in general. For instance, from CGEL page 1372:
[1.i] Most of it(i) she had written ____(i) herself.
[1.ii] Anything you don't eat(i) put ____(i) back in the fridge.
[1.iii] It appears that from one of them(i) he had borrowed several hundred dollars ____(i).
Your question would also apply to those examples too -- as to how the reader processes the fronted element and its associated gap, whether the info is processed once or twice.
I'm not sure where that kind of info can be found. Perhaps somewhere in some subfield within linguistics, maybe?
My questions here is, is this word, ‘where,’ interpreted twice? I mean, does ‘where’ have its semantic role in both ‘where do you think’ and ‘he went Ø afterwards’? Or just once?
For your example, it seems to me that the word "where" doesn't mean much semantically until the embedded content clause is actually getting processed or has been processed. That means the associated gap has to be processed first in order for the reader to know where the word "where" fits semantically in that sentence. But this is just me thinking out loud.