Well, for those already flying to Midway, it shouldn't mean much in direct costs. Most tenants have signed agreements with the airport that limit the rates that they can be charged to fly from the airport. For passengers, it's a different story. Retail and parking rates could rise (especially since they can't charge airlines any more), but that's not what we're talking about here. The place where we could see the most change is in airlines looking to serve Chicago.
Now that Midway and O'Hare will be owned and operated by different entities, it could make them more aggressive in pursuing new carriers and new service. If an airline wants to begin flying to Chicago, you know that Midway will want to do what they can to wrestle that airline away from O'Hare. Of course, there's a limit on what they can do. They can't offer preferential rates to one airline, as far as I know, but they can work on some incentive packages and at the very least, be more accommodating when an airline is looking for gate space.
The country will be watching this very closely. Midway's privatization if part of a federal test program, and Midway is the only large airport taking part. So, if this works well, we can expect to see it spread to other airports throughout the US, especially since it's a nice way to get a big chunk of change.