Rental counters are on both sides.
Here's the thing to keep in mind, and not just the OP but everyone here....DME is not set up to enable guests to tailor it to their specific needs. We are seeing more and more questions here asking how to get DME to drop at a resort other than the guest's resort, or at a park, or picking a guest up at a park or an alternative resort to enable them to spend more time in a park on departure day. Many different scenarios keep coming up.
DME takes MCO arrivals, from the airport to the guest's resort. Then, DME picks the guest up, at their resort 3 hrs before flight departure, and takes them to the appropriate side of MCO for their particular airline.