I have a good working relationship with two of the schedulers for certain shops, and I am able to email and request bundles of shops for specific dates and times (though not before they hit the job board). I usually am granted the locations I request. But no, no self-assign privileges. I wish.
I will sometimes make a note about a shop that I'm applying for saying, "Please only assign if I can also get xxx shop." or "Please only assign if I can get at least 4-5 shops in area I've applied for due to distance. Please do not assign one or two." They can then check to see if it was assigned or co-ordinate with the other scheduler. It has always worked well when I'm going a distance.