DL award tickets on United come out of the same inventory pool as United awards. Premiers do not get any kind of priority when first requesting free award travel. It's first come, first served. Premier status does weigh heavily if you are waitlisted however.

Pattius is correct that Delta agents sometimes have difficulty navigating United's system looking for award inventory. It is also correct that Delta must do the ticketing. But you can call United directly to find you a qualifing itineray (but not book it) on United. You can then call Delta back and give them the info and they can ticket it for you.

Beckles had this problem when he was trying unsuccesfully to use Delta miles to get onboard our United flight to the Party In Paradise. He contacted me and I was able to get a United agent to find a qualifiing itineray and he just called Delta with the info and then got it with no problem.