Although we haven't seen them, a lot of incredibly convoluted events must have taken place for the Mirror Universe characters to continually be in the same places as their Primeverse counterparts.
When all of humanity was enslaved by the Klingons and Cardassians a century ago yet the DS9 crew's counterparts are all interacting on and around the MU version of DS9 is at least
as fishy as Kirk bumping into Scotty bumping into Old Spock who just happens to have a way to get them both back to the Enterprise.
I go with what Spock said in "City on the Edge of Forever"...
SPOCK: First, I believe we have about a week before McCoy arrives, but we can't be certain.
KIRK: Arrives where? Honolulu, Boise, San Diego? Why not Outer Mongolia, for that matter?
SPOCK: There is a theory. There could be some logic to the belief that time is fluid, like a river, with currents, eddies, backwash.
KIRK: And the same currents that swept McCoy to a certain time and place might sweep us there, too.
SPOCK: Unless that is true, Captain, we have no hope.