I don't know whether this is true, but I think I've heard it said in connection with this story.
Egypt had a solar based-calendar, so (to a decent approximation) on every named date the sun would be in the same place in the sky.
So all that would be needed was for it be known that the sun shined straight down the well on (for instance) the 14th day of the 2nd Month of Growth (I had to look up the Egyptian calendar to get that date!), and Eratosthenes just needed to measure on that date.
The trickiest part was to know the effective distance component in the North/South direction. Luckily the Nile is roughly oriented that way anyway so the distance between the two cities is roughly the same as the North/South component
The communication requirements are minimal. The only real requirement is that we agree on which day to perform the measurement.
You and I could sit at Christmas dinner, and conspire to measure the shadow of a 1metre pole at solar noon on Easter Sunday, at our respective locations.
We could then compare those measurements when we meet next Christmas.
Egypt had a solar based-calendar, so (to a decent approximation) on every named date the sun would be in the same place in the sky.
So all that would be needed was for it be known that the sun shined straight down the well on (for instance) the 14th day of the 2nd Month of Growth (I had to look up the Egyptian calendar to get that date!), and Eratosthenes just needed to measure on that date.