When you are on Jury service, you are only ever seeing one specific sequence of posts, and it is always the sequence which leads from the OP directly to the alerted post. All other "branches" of the conversation are removed.
The Jury page always begins with the OP and ends with the alerted post. In between those two posts will be every post which is a DIRECT reply in the chain. Sometimes there might not be any posts between them -- the alerted post might be a direct reply to the OP, in which case you would just see the OP and then the alerted post. But sometimes there might be a long chain of direct replies in between the two, and they could have been posted by multiple people.
By way of an example, here's a short sequence from a thread I posted yesterday:
https://www.democraticunderground.com/101314113#post15
At post 15 you can see a reply to the OP by BigmanPigman. Post 15 gets a response from Attilatheblond (post 16), and then Attilatheblond gets a response from BigmanPigman (post 21).
But post 15 *also* gets a response from TrueBlueAmerican (post 31). Now, if TrueBlueAmerican's post were to be alerted on, then posts 16 and 21 are irrelevant because they're not directly part of the same conversation. They don't provide any additional context, and so they do not need to be provided to the Jury.
Instead, the Jury would just see the direct sequence of events: The OP, then post 15, then post 31 (the alerted post).
The bottom line is that when you serve on a Jury you are only going to see the OP, the alerted post, and then any DIRECT replies that link those two posts together. Everything else is stripped out. I hope that makes makes sense.