When a player that is riding a horse, rides out of players 'player detection range' (something like entity-tracking-range in server properties),
his character will teleport to a random position, leaving the horse without the rider (that's what is showing for other players).

The rider needs to step down from the horse near a player to be properly shown for others.

This also happens, when someone will ride into our detection range on a horse. We won't be able to see them until they get off the horse.

Here I'm leaving a link to a YouTube viedo that I made to show what I mean, in case my words fail to pass the message.

https://youtu.be/nNQb3_JtXeo