In fact there is a good chance they would have only 1 9th-great-grand-parent in common, because of ancestors who were married more than once and had children by different spouses. In either case, 10th cousins may not share any DNA, unless they are descended in the direct male line from a common ancestor and share the same Y chromosome.
Normally second cousins would share 2 of 8 great-grandparents in common. In my case I have second cousins with whom I have only one great-grandparent in common, since they are descended from my grandfather's stepmother.
Obama’s family tree looks like a spiderweb.
Almost, but not quite. They also could share mtDNA, if both descend in an unbroken female-to-female line from a common 9th-great-grandmother.
Moreover, total genome "sample scans" sold by the firms 23andMe and deCODEme now are sometimes able to pick up shared DNA as far as eight generations back, no matter that the ancestral trails involved are mixed between males and females. For example, my own 23andMe DNA results show a "haploblock in common" with a known sixth cousin -- whose paper trail back to our common 5th-great-grandfather passes thru several female ancestors.
Now to be sure, these total genome scans (based BTW on the Illumina chip technology) aren't yet as reliable for genealogy as are the standard Y-chromosome and mtDNA tests. But when the cost of total genome sequencing -- as opposed to today's "sampling" -- falls down to ca. $100.00 per shot, then we might be able to do reliable DNA tracing back to the generation of our 9th-great-grandparents. I guess we'll just have to be patient for another five or ten years!