Imagine yourself – a man – sitting next to someone you want to talk to – a woman – on a bus, train, plane, etc. After a while you say: 'One of us has to start talking / break the ice, don't they? / doesn't he? / doesn't she?'
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It seems a little strange, but that's because people wouldn't phrase the (rhetorical) question that way.
Singular they is the modern, gender neutral term used increasingly in situations of mixed-gender crowds.
It's even more appropriate in the case where the mixed-'crowd' contains only two people. Using he would seem to put the onus of talking on the man, and using she would do the same for the women. So using the gender neutral singular they works best.
For more info on the use of singular they, see this wiki article. It seems to be acceptable in most places, except some American audiences where it will 'make people doubt your literacy', but in my experience this seems to be vanishingly applicable in the modern sense.
ETA (because I jumped the gun and didn't think the whole thing through, before answering the call for gender-neutral singular) in the particular case mentioned by the OP, the pronoun should indeed by we, as pointed out by a whole lot of people in the comment threads as well as by Matt in the answer below.
I'd rephrase it entirely, as something like, "One of us has to start talking, don't you think?"