According to the EU novel DS9: Warpath, the multi-dimensional transport was designed by Smiley O'Brien, presumably using a mixture of info gleaned from reports about James Kirk's travel to the mirror universe, his own knowledge of transporter technology and scans of the technique used by Kira and Bashir to return to their own universe at the end of DS9: Crossover
The compliment made him pause. “Thank you, Intendant.”
If only that fool Smiley could see what I’ve made of his little toy,
she thought. What a pity he won’t live long enough to witness its true
This is confirmed in the Official Star Trek Fact File #59
One of the greatest problems faced by the Terrans in retrieving Sisko
from Deep Space Nine is in recreating the transporter malfunction that
allows travel between the realities. Miles 'Smiley' O'Brien of the
mirror universe is at first reluctant to help Kira Nerys and Bashir to
escape from his world, but later develops a remote transporter unit
that can duplicate exactly the effect on a Cardassian-designed
transporter as the two previous accidental set of circumstances. In
addition to this complex engineering, the remote transporter unit
negates the need for an operator on either side of the universes'
divide to initiate transportation of one or more individuals, thus
allowing the user to act with a great deal of flexibility.
As to how Rom got the schematics, that's pretty simple to answer. He was given them by Chief O'Brien...
ROM: My engineering notebook? (realizing) With the schematics for the multi-dimensional transporter that Chief O'Brien gave me to study. (a beat) You stole it...
DS9: The Emperor's New Cloak - Original Screenplay
... , who presumably scanned the device that was left for Captain Sisko by (alt)Jennifer Sisko in DS9: Shattered Mirror a couple of seasons earlier.
Sisko, O'BRIEN, Kira and DAX are gathered around the Ops table. O'Brien is evaluating the (TECH) device Jennifer left Sisko. He's opened a small panel on the device and is poking its innards.
O'BRIEN: You're sure this is what Smiley used to transport you to the alternate universe?
SISKO: Looks like it.
DS9: Shattered Mirror - Original Screenplay