Translation:The lawyers are watching the judges, and we are watching the lawyers.
Oh, I am watching you. Now that would be "figyelni". To pay special attention, in order to find out something about you. When you watch tv, that is just "nézni". But when you keep an eye on the tv to catch the weather forecast, that is "figyelni".
There is an important difference between "figyelni valakiRE" and "figyelni valakiT". This is for people. For objects, the difference is not as important.
"Az ügyvéd figyeli a bírót" - "The lawyer is watching the judge". - This is a little unnerving if he has bad intentions.
"Az ügyvéd (oda)figyel a bíróra" - "The lawyer is paying attention to the judge". - That is very smart of him.
So, those people up there in the sentence are not paying attention to what the others are saying. No, they are keeping an eye on them.
"Figyelni" can also be "to observe". Especially "megfigyelni".
"Watch and learn" - "Figyelj és tanulj"
"We are observing your Earth" - "Megfigyeljük az önök Földjét"
"The Observer" - "A (meg)figyelő"
"I observe these animals" - "Megfigyelem ezeket az állatokat"
"J" and "ly" are letters of the Hungarian alphabet. They do sound the same. "Lj" are just two letters that happened to be next to each other in a word. You are welcome to try to properly pronounce them until your tongue gets the cramps. But here they want to morph into "jj". They are better pronounced in other words, for example, in "feljön".
I can compare this to what happens to "n" in an "ng" situation. DuoliNGo, baNG, oNGoing. It is just the natural laziness of our speech organs.
Well, it might be dependent of each speaker's native language, but I don't feel uncomfortable to pronounce LJ as two different sound, one after the other. In my native Ukrainian figyeljük sounds фідєльюк :) Therefore, such an aspect is a clue to correct Hungarian pronounciation.
There was a Hungarian politician who famously spelled out the individual sounds in such situations. That was his style. Maybe it was also a clue to his roots, I don't know.
Anyway, these morphings just happen naturally. And I am sure you are right: what happens naturally to you depends on your native language.
And I think most people are not aware of the fact that these morphings happen when they speak. So if you ask them if there is such a rule, they will be surprised. So, maybe the best option is to listen and pay attention (figyelni) to how the natives speak and try to get as close as you can with your own pronunciation.