Hey guys, for whats its worth i have been battling trying to get my air drop to work on my 2013 New Mac Pro. It works between my phone and others phone, as well as between my phone and my mac book air.
I then figured out that the minute i go into system preferences and log out of iCloud it would instantly start working. If the airdrop window was in the background, the sec it finishes the log out all my devices pop up in air drop. So after trying to uncheck one by one all the iCloud services to see if there was some kind of conflict I finally found a solution.
I actually had to log out of iCloud on my mac book air, and re log in. after i did that i was able to log back into iCloud on the mac pro and now its all working great. What tipped me off is that on my phone and mac pro (with iCloud logged out) my mac book air would show up but the computer name was off.. it was displaying one of my email addresses instead... so after logging that machine out and back in I'm good to go.
moral of the story... i thing a lot of the issues have to due with the iCloud accounts across all devices. hope that helps someone.