To stick an oar in, and starting with "remember that the rules for duplicated missions have changed over the years, so don't be surprised that things have become a bit clunky"...
I think the key point is to remember that you always are looking at your side of the mission. If you seed it, it's the "big side", otherwise it's the "opponent's side".
(Yes, I'm ignoring all the weird "spin the mission" mechanics.)
When you duplicate a mission, you get this weird welding effect - we treat it as a single mission, but each of you look at your copy only. The root reason is that a bunch of rules and mechanics (depending on format) say "you can only attempt your missions/missions you seeded". So the rules say "great, you both seeded it, so both of you can treat it as your mission and ignore the other side".
To use an example, think of
Compromised Mission - if you both seed this, both of you are seeing the Rom/Kli side, and importantly: neither of you are seeing the Fed side. You can't interact with the opponent's side of the mission, nor they with yours.
Which is where we get to Access Denied - it's looking for an objective targeting your mission. But it's not - they're targeting *their* mission, because they literally can't see your side of the mission. They couldn't target the "opponent's mission" there if they wanted to.
Now, stepping back to the why and rationale - the purpose of this whole mechanic is so you don't get punished by duplicating a mission with your opponent. (The old rule was that the 2nd player had to swap out for a universal mission, which made many players and TDs cranky because someone had to keep a stash of replacement missions handy).
(Previous to *that*, you'd both just seed your mission separately, which meant you could have two Earths - once the game started caring about regions and homeworlds and such that wasn't feasible anymore).
So, to bring back around - the current rule is based around "if you both seeded this mission, both of you can treat it as yours so neither of you get punished". The flip side is that cards like Access Denied don't work, because we don't want your opponent to get punished for seeding their mission either just because you happened to seed the same one.