Here are a couple of spacetime diagrams. These are essentially displacement-versus-time diagrams, except that conventionally time is shown up the page and position horizontally. Also, the scale is picked to be seconds and light seconds, so that the speed of light (one light second per second) gives the paths followed by light a slope of ##\pm 1##.
Apologies for the hand-sketching. All lines are meant to be straight. I can draw computer-generated versions at the weekend if it helps.
Here is the first diagram:
View attachment 244704
In this diagram there is one observer, marked in red, stationary at the origin. A second observer, marked in blue, passes by moving to the right. At the instant they pass, a flash of light is emitted, marked in fine orange lines. Because the slope of these lines is ##\pm 1##, the midpoint ("where the flash was emitted") is always where the red observer is. The blue observer is to the right of this at all times after the actual emission event.
Here's the same scenario in the blue observer's rest frame:
View attachment 244703
Now the blue observer is stationary at the origin and the red observer approaches moving to the left. Again the flashes are emitted when the observers pass, but this time the midpoint of the flashes is where the blue observer is.
So in both cases "where the flash was emitted" is the origin of the spatial coordinates (##x=0## in one case, ##x'=0## in the other). But it's clear that the two frames mean different things by this - ##x=0## is where the red observer is and ##x'=0## is where the blue observer is. And they are not in the same place.
If, as
@Nugatory has suggested a couple of times, you actually mean that "the flash happens when and where the observers pass one another" then everyone will agree. But this is an event (a place and time), not a point. A point in space has multiple different meanings depending on which frame is doing the describing.