We're a small startup, and we don't have a really structured process. Also I have been the only person asking a detailed coding question (in consultation with my boss). Everybody else on the team has been responsible for those other questions.
I should also add that I personally prefer to do coding questions in person on a whiteboard, rather than doing them during a phone screen with some sort of collaboration software.
My technique of explaining the algorithm before asking the interviewee to implement it works way better with both people in front of a whiteboard.
I'm not saying everybody should do it this way. I'm just saying that this is my current approach and it seems like it's working OK.