Basically, it works like it worked in RSE.
First off, you have to make a Coordinator Stats with the pokemon you're going to use in the contests. You can put all your mons, or just a couple, it's up to you.
To start a contest, go to the coordinator station and post the Contest Attribute you're looking for, or look if there's a contest that you like. When 4 people join a contest, a thread can be started.
After starting the thread(with the name of who's partecipating and possibly the rank/attribute of the contest), you'll have to wait for a judge to start the judging. Actually, there are only 2 aviable, me and ATF. But don't start rushing the judges unless they're really late.