I often fly on small regional jets. In my experience, they have a good idea of when the bins will be full and are proactive about it. They'll also know how much space is left as people board, and sometimes people at the back of the line gate check the bag before boarding, even if they weren't asking for gate checked bags prior to boarding.
When they take your bag at the gate, there are two options for what happens to it. You'll need to know which option they are using - they'll tell you.
1 - Sometimes it is checked through to your destination. They will add that white barcode tag to it at the gate and tell you to pick it up at baggage claim at your destination.
2 - Sometimes they have "a la carte" service, where you deposit your bag at the end of the jetway and then claim it on the jetway at your arrival. They will give you a tag for these bags - it's usually neon pink or green. If your plane is too small to use a jetway, they'll have a rolling cart set up next to the plane stairs and you put your bag on it and pick it up when you disembark the plane.
Regardless, as others have noted, DME is not doing the luggage service at this time.