When using /tellraw's insertion tag on a server, nothing happens.
While on server:

/tellraw @a {"text":"Hello!","insertion":"Hello there."}

Shift + Click does nothing

While in single player:

/tellraw @a {"text":"Hello!","insertion":"Hello there."}

Shift + Click works.

In fact, it works on L.A.N. worlds! So I don't have any idea of how the server is a problem.

Marcono claims to know the reason.