You are correct, they are writing in a notebook. This is a common practice at events, and especially in Best of 3 rounds, so that you can know what your opponent has in his/her arsenal and can make more accurate predictions. Common things to write down include:
• Pokémon on opposing team.
• Moves, Abilities, likely natures.
• If your Pokémon outspeeds theirs or vice versa.
• Specific details. For example, Mega Gengar's Sludge Bomb does just enough damage that Primal Kyogre can't OHKO with Water Spout.
If you need to know more let me know :)
Source: Experience in VGC and watching streams.