Sorry for answering my own question, but I finally got some insights on the topic.
1) Where I can find those "released documents"?
According to Marco Langbroek's (the same amateur satellite tracker that took the image in the OP) excellent review on the topic:
On September 6, 2016, The Intercept published a number of new
documents from the Snowden files, in the context of an inquiry into
“targeted killings” (drone strikes) and the role and function in these
of Menwith Hill, a large US-British SIGINT base in Yorkshire, UK
The actual document where the eavesdropping of Mentor-4 on Thuraya-2 is mentioned can be found here:
The context for this specific release is reviewed in this publication of The Intercept.
2) What was the minimum distance (or the average distance) between Mentor-4 and Thuraya-2 during the eavesdropping?
According to amateur observations, Mentor-4 moved close to Thuraya-2 just as Mentor-2 moved out from the region. The leaked documents and these observations made clear that the eavesdropping of Mentor-4 was a continuation of the activities of another satellite (which now we know was Mentor-2) on Thuraya-2, but the main difference amateur observers spotted was that Mentor-4 got extremely close to Thuraya-2 (way more than its predecessor apparently).
So I'm going to calculate the distance between both objects using TLE's around the date this of this photograph
Which was taken in May 2016, apparently. I can retrive a TLE for Thuraya-2 from space-track.org for the 15 of May, 2016, but I can't do it for Mentor-4 (unsurprisingly since this database is controled by the US government and the satellite operations are still classified). For Mentor-4 TLEs I turn to amateur observers. Mike McCant did an excellent job and there's a TLE for Mentor-4 available also for the 15 of May, 2016. Using both in HeavenSat, I can spot their distances from a specific location on Earth and their coordinates to calculate their actual distance.
At 19:14:37 GMT, Mentor-4 and Thuraya-2 had an angular separation of just 3.35 arcseconds, which (using the distance values available from the TLEs) corresponds just to 31 km between both objects!.
Considering the fact that Mentor-4 is believed to have a 100 m dish antenna, this means that, at that time, it was so close that an observer from Thuraya-2 could have seen Mentor's-4 dish ~37% the size of the full Moon (certainly enought to spot details of the other satellite with the naked eye). I think the distance might have been quite constant between both but maybe there are even closer values after so many years of chasing each other.
3) How does a satellite eavesdrop on another and why it needs to be close to it to do that?
Here I would accept as a plausible explanation the one made by user3528438. The radio link between Thuraya-2 and ground stations can't be economically feasible using omnidirectional antennas and must be focused towards the target in GEO. That means that there is a relatively narrow beam where the information is traveling. I guess that getting closer to the center of the beam means getting a higher amplitude on that signal. I guess that being 30 km away from the center of the beam with a 100 m dish antenna is enought to pick the signal, so the beam can't be as collimated as a laser.