How do observers communicate their respective times in Einstein synchronization procedure?

Question

This is in reference to the synchronization condition detailed in Einstein's 1905 paper:

If at the point $A$ of space there is a clock, an observer at $A$ can determine the time values of events in the immediate proximity of $A$ by finding the positions of the hands which are simultaneous with these events. If there is at the point $B$ of space another clock in all respects resembling the one at $A$, it is possible for an observer at $B$ to determine the time values of events in the immediate neighbourhood of $B$. But it is not possible without further assumption to compare, in respect of time, an event at $A$ with an event at $B$. We have so far defined only an “$A$ time” and a “$B$ time.” We have not defined a common “time” for $A$ and $B$, for the latter cannot be defined at all unless we establish by definition that the “time” required by light to travel from $A$ to $B$ equals the “time” it requires to travel from $B$ to $A$. Let a ray of light start at the “$A$ time” $t_A$ from $A$ towards $B$, let it at the “$B$ time” $t_B$ be reflected at $B$ in the direction of $A$, and arrive again at $A$ at the “$A$ time” $t^′_A$.

In accordance with definition the two clocks synchronize if

$t_B − t_A = t^′_A − t_B$.

If $B$ is moving at a constant velocity w.r.t. $A$, and if $A$ wants to perform a synchronization check, how does he find out the value of $t_B$ ? Does he measure it himself or does $B$ communicate the value of $t_B$ to $A$? If $A$ measures $t_B$ himself, could a synchronization check performed by $B$ yield a different result? (since then $B$ would have to measure "$A$ time")

Answer 1

"Special relativity statements involving an "observer" are in some measure articulating a similar kind of impersonal relative direction. An "observer" is a perspective in that it is a context from which events in other inertial reference frames are evaluated but it is not the sort of perspective that a single particular person would have: it is not localized and it is not associated with a particular point in space, but rather with an entire inertial reference frame that may exist anywhere in the universe (given certain lengthy mathematical specifications and caveats.)" https://en.wikipedia.org/wiki/Observer_(special_relativity)

Clocks A and B create so -called observer's rest frame. Observer synchronizes these clocks by means of Einstein synchrony convention (velocity of light back and forth is c).

https://en.wikipedia.org/wiki/Einstein_synchronisation

Every observer conducts measurements by means of synchronized clocks. Observer needs two synchronized clocks at least in his frame. If observer measures dilation of moving clock, he first compares readings of moving clock with the first clock A (in immediate vicinity) and some later with the second clock B. Single clock measures shorter time interval than two spatially separated clocks.

https://arxiv.org/ftp/physics/papers/0512/0512013.pdf

Einstein synchronization is only a special case of a more broader synchronization scheme. Reichenbach's synchronization leaves the two-way speed of light invariant, but allows for different one-way speeds.

https://en.wikipedia.org/wiki/One-way_speed_of_light

Answer 2

Although Einstein did not explicitly mention, $B$ sends back the information of $t_B$ to $A$. Then $A$ can confirm that $A$'s clock is synchronized to $B$'s. To convince $B$ that $B$'s clock is synchronized to $A$'s, $A$ has to send the information of $t_{A2}$ to $B$.

This operational view has an advantage that all the process can be experimentally verifiable in principle, while the introduction of the third party other than $A$ and $B$ does not have.