You generally do not need to remove the engine to change the transmission on a 1998 Ford Mystique. While it can be a challenging job, experienced mechanics usually perform this repair by dropping the transmission from underneath the car, after disconnecting it from the engine and other components. However, access might be tighter than on some other vehicles.
That said, some shops *might* choose to remove the engine to make the transmission replacement easier, depending on their preference and the specific circumstances. This is not a typical approach, though. Ask the mechanic what their planned procedure is if you are getting the work done professionally.