Starting from the incident from "Mirror, Mirror", the mirror universe terran empire/memory omega managed to create the interuniverse teleportation technology.
The federation, despite having vastly more resources than memory omega, failed to do so - it even failed when having the DS9 interuniverse teleporter device. Highly unimpressive.
Actually MM had nothing to do the the crossover gizmo. It didn't come into play until the second DS9 MU episode, Shattered Mirror. And it was used by Smiley who didn't even know about Memory Omega at the time, so I doubt they had anything to do with it. I'm pretty sure the Federation never actually used it themselves, so as far as I know they didn't have any access to it themselves.
JD, I didn't intend to continue the conversation on this topic in this thread, but...what you said is directly contradicted by canon: DS9 - The emperor's new cloak.
By the timeframe of that episode, the federation had complete plans of the "multidimensional transporter" (courtesy of the mirror universe) - "multi" also carrying certain implications pertaining to the capabilities of said device.