The Paths do not share vision with each other. They are dead corpses reanimated with chakra. They can't share anything ... with each other. The only sharing going on is with the Paths and Nagato himself, and seeing how he is able to hold conversations through his Paths with other people, it is a safe bet to say that he can hear what his Paths hear as well as see what they see.
Yet the only Paths caught in Jiraiya's sound genjutsu were the 3 sent directly after him.
That proves a couple of things to me: the most important of which is that Nagato himself does not fall under any genjutsu that the Paths do. And why should he? Genjutsu takes over the chakra in the mind of the victim; the minds in this case being those of the Paths, whose chakra networks are reactivated by the chakra Nagato is broadcasting. Look at the description of what
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actually does to the opponent. You think Nagato would still be able to broadcast the chakra needed to talk through his Paths, as well as mobilise the other Paths, if his nerves and mind were paralyzed?
Then there's also the fact that the
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. Literally the only person who was able to trace Pain's chakra back to his source was Naruto in Sage Mode, and that was by stabbing himself with one of his chakra rods. By that logic (and the other fact that Nagato did not turn into stone as well when Preta Path did), it is apparent that Nagato does not receive any chakra back from the Paths, thus the chakra in his mind is not at risk of being taken over via his Paths as proxy, and thus he is not in danger of falling for the same genjutsu as his Paths.
With that said, Itachi is not likely to catch a Path in a genjutsu when 6+ pairs of eyes can study his movements quite well without needing to look in his eyes, whereas Animal Path summons could take the risk of looking in his eyes due to the fact that they can be dispelled easily. Even if one falls to genjutsu, Tsukuyomi is doing nothing to a Path/animal besides wasting chakra, Pain can break that Path/animal out of a non-Tsukuyomi genjutsu by simply
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, and Nagato has
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with
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himself, meaning he could very likely break finger genjutsu/weaker Sharingan genjutsu through his Paths.
Amaterasu gets absorbed with Preta Path, countered by Deva Path, or outright
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by an Animal Path summon, and burns slowly enough that Preta can just run up to another Path and absorb it. And Itachi will already start showing signs of fatigue if he has to use both Amaterasu and Tsukuyomi in the same match. Animal and Asura Path could just literally overwhelm Itachi with summons and missiles, and a routine assist from Deva/Preta until he dies.