On one hand, you're both correct... but on the other hand, it seems like you're ALSO freely handing out the info that the Federation could potentially be destroyed at any time by monkeying around with the transporters in certain ways and bringing through an MU invasion force. Not exactly something you want cadets discussing at the bar with agents of Federation adversaries potentially sitting at nearby tables - or that you would want a disgruntled ex-cadet like Nick Locarno knowing about.
Maybe they tell them about the parallel universes, but leave the technical details classified?
A civilization could be destroyed at any time by an asteroid or a supernova or a plague. That doesn't mean you make it illegal to study them. On the contrary, hiding information is absolutely the worst, stupidest, most self-defeating way to deal with a potential threat. People can't defend against it if they don't know about it.
Besides, there are a couple of things we know from "Mirror, Mirror." One is that the crossover was a freak event occurring because of a rare weakening in the dimensional "barrier" between universes caused by the ion storm. It wasn't the transporters alone that made it possible.
The other, more important thing is that
the characters already had a theoretical understanding of the situation. Kirk was able to deduce what was going on and how it had happened before he even consulted the computer. The idea of a parallel universe wasn't some shocking, unprecedented idea, but something that a non-scientist could recognize and accept easily. The mirror computer
already had the necessary information to tell Scotty how to recreate the accident. Kirk and Scott already had the theoretical knowledge to speak handily about the "field density between universes" (whatever that's supposed to mean). Understanding the situation came easily; their obstacle was merely one of dealing with the Mirror crew long enough to carry out an otherwise routine procedure. That tells us that 23rd-century science already had sufficient theoretical knowledge to take a visit to a parallel universe in stride.
So, really, what is there to keep secret? Federation science as of "Mirror, Mirror" clearly already understands that parallel universes can exist (naturally, since the theory would be centuries old by that point) and grasps the physics and engineering principles behind a crossover. The discovery of the Mirror Universe's existence and a mechanism for making the crossing would merely be confirmation of existing theory.