I'd agree with the comments that the same rules legally apply when you're on a bicycle as if you were driving. However, my understanding is that the stop sign means "Stop, and proceed when it is safe" - I'm not aware of any legal obligation to take turns. This was summarized as "Actually, whoever goes first has the right of way." (since it is no longer safe for others to go, and it would be foolish for them to go). Drivers generally take turns since it works better than constant contests to see who goes first.
On a bicycle, the key is traffic and location. If I'm in the city with 2 one lane streets (i.e. one way), there is only 1 other driver. If he waves me on, I figure it is safe so I accept it. If I'm trying to cross a 4 lanes of traffic (or sometimes even 2), I may wait if I'm not convinced drivers in other lane will wait just because the closest driver waived me on.