Ancestry doesn't provide a way to do this yet, but hopefully they will, since similar tools are available elsewhere.
When Ancestry releases the raw data, you will be able to upload it to GEDMatch.com and compare it to others who have uploaded to GEDMatch... the trick will "just" be getting all of these matches to also upload to GEDMatch. Ha! (Hence the hope that Ancestry will offer this in-house.)
At GEDMatch you can compare both DNA of any two people (not just yourself) for common segments and GEDCOMs for common ancestor names, so long as you have the ID numbers for the other people's stuff. (That's why I put mine in my profile here.)
So, if a few hardworking volunteers can make this happen for a significant number of genealogists, and not even charge a penny, I think Ancestry should be able to swing it. ;) (Use that Feedback button if you agree!) Plus, Ancestry could probably improve such a feature to find common names throughout the tree (not just ancestors), like you said, kind of like how it currently works with the leaf hints.
Again, let them know if this is important to you. In the past, Ancestry has sort of implied that this kind of feature is not a "mainstream" interest. Tell them you feel otherwise! :)