As far as I could tell, this isn't possible, there's even a similar post and the solutions there are
I have a potential solution, it's going to be tedious, but it might just work.
This assumes all your paths are relatively simple paths (puzzle pieces) - if they are rather complicated this whole thing is probably not worth it.
Also, I would recommend working in outline mode for this.
Select all your paths and use pathfinder > outline.
Then ungroup everything and select the outer paths and cut them to a new layer (Ctrl+X then Ctrl+F on new layer).
Now lock the layer which has all your loose paths, you will get back to those.
Using the Direct Selection Tool (A) carefully select the insides of the shapes and delete all of them.
Once you've done this, go to your other layer with loose paths and you can use the standard selection tool (V) to click on each of the paths on top and delete them.
After you've cleaned everything up you can select all the inner shapes and Join them (Ctrl+J)
Your end result should be something along the lines of individual paths.
Pathfinder doesn't do the best job and you might have to not carefully extract certain parts of the main path. (I had to in this example)
In any case, I would recommend recreating it for best accuracy.