The most frightening thing is for two pilots to announce that they are "overhead" at the same time because by definition, iaw the standard procedure, they will be in exactly the same location at the same altitude. If the other aircraft isn't seen, it leaves at least one, or both pilots in a quandary - do you carry on with the procedure, do you maintain altitude, or what?
If you are doing a proper standard join you don't actually fly precisely "overhead".
This is the way I teach it:-
Imagine a bicycle wheel with the axle centred on the middle of the airfield/runway. The rim of the wheel encompasses the boundary of the airfield (or the ends of the runway).
When you are visual with the field (assuming a left hand circuit) turn the a/c to place the airfield on the left (about ten o'clock) and now fly around the rim (notice this will not place the a/c overhead the airfield but a short distance away from the zenith at the axle) until the threshold of the runway on which you intend to land is at nine o'clock (in a low wing out of sight under the left wing).
You are now proceeding towards the dead side and can start descent (below 2,000 ft) still following the rim of the wheel as it is safer to descend turning as a) you are clearing airspace below you and b) able to see traffic taking off or going around and you can intelligently decide where to fit into the pattern.
From certain directions this means it will take a little longer to join the circuit. The difficulty arises when other airspace users decide to "short circuit" the procedure if they are already approaching from the deadside. Eg approaching from the north for RW 27 with a left hand circuit.
Essentially what we are discussing is a non radio join in which case you would need to look at the Signals Square.
For me the term
Overhead Join is a misnomer because if flown correctly you are never actually overhead the airfield but on the perimeter.
Logically if all a/c follow the proper full joining procedure whoever calls "Deadside Descending" is ahead of other joining who have yet to make that call.
That said some of the joins which I see the occasional aircraft perform are far from standard so it's essential to maintain a good lookout at all times.