Will self funding a PhD hurt employment chances? I am thinking of self funding a PhD in Applied Mathematics at either the University of Oxford or Cambridge. Would employers, both for academic (postdoc etc) and industry (research) jobs think less of a PhD holder if he or she is self funded?

E.g. Would they feel that because I failed to secure funding for their PhD, I am inferior to students who did? Although I am planning to self fund to UK schools, it would be nice to hear about views from across the pond (USA) as well.
There's nothing on your CV that needs to indicate exactly how you were supported during your Ph.D. In fact, if you were self-funded, and managed to get an occasional fellowship or other form of support, that's a plus !

All a recruiting committee should (and does) care about post-PhD is the quality of your work (for faculty positions there are additional issues). No one cares about how you were paid to do that work.