One needs to follow expected courtesies. The law in my state at least is vague. If one needs to wait until clear and "the gun" goes off at the same time, this contradicts itself. I am assuming both myself and the oncoming driver are on a secondary road, crossing a primary road. (On a 4-way, I consider a straight shot to take priority over a left turn. I also "pair off" going straight with a straight shot coming at me in the interest of keeping traffic moving, even if it isn't my turn.)
If the cross traffic was in the opposite lane I'd go first, as I had "the jump" on it having cleared. If it was in my lane I'd let the other guy go.
That all gets thrown out the window when the oncoming driver just waves b/c he doesn't want the drama.
I almost clocked a guy head on in this scenario: Busy 5 lane road (2 lanes each way and a middle left turn) in a commercial/ burger joint strip of road. We both finally got clear and felt we could go if we gunned it. Narrowly missed a wreck.
Code:
7. Stop signs. Unless directed to proceed by a law enforcement
officer or traffic control device, an operator of a vehicle
approaching a stop sign shall stop and:
A. Yield the right-of-way to a vehicle that has entered the
intersection or that is approaching so closely as to constitute an
immediate hazard; and [1993, c. 683, Pt. A, §2 (NEW); 1993, c.
683, Pt. B, §5 (AFF).]
B. Having yielded, an operator may proceed. All other operators
approaching the intersection shall yield the right-of-way to the
vehicle so proceeding. [1993, c. 683, Pt. A, §2 (NEW); 1993, c.
683, Pt. B, §5 (AFF).]
[ 1993, c. 683, Pt. A, §2 (NEW); 1993, c. 683, Pt. B, §5 (AFF) .
http://www.mainelegislature.org/legis/statutes/29-A/title29-Asec2057.html