They're playing Trials.
No, just kidding. I'm not sure I would be able to tell -- the only time it would be obvious would be predictable "good connection -- oh wait, now they're teleporting everywhere when I get the drop on them -- and now they're back to good connections." Otherwise, I think that's something Bungie will have to investigate. And as for headshots not registering, that is more likely lag than active cheating -- it looked like a good kill to you; on their end, they'd already moved, but it hadn't updated your machine yet.
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Yeah that makes sense. I've seen it before, but not as obvious as last night. Still feels odd, but lag could be the reason.