Changes like these made on the client will not replicate to the server (although there are some exceptions, none apply to your case). You would need to use a RemoteEvent to achieve this.
If you want short example:
Put Remote Event into ReplicatedStorage and give it name ChangePlayerTextBoxText
LocalScript:
local ReplicatedStorage = game:GetService("ReplicatedStorage")
local RemoteEvent = ReplicatedStorage:WaitForChild("ChangePlayerTextBoxText")
local Test = game.Players.LocalPlayer.ScreenGui.Frame.TextBox.Text
RemoteEvent:FireServer(Test)--this will fire RemoteEvent on server with think called Test like you will be able to use it in basic script--
Script:
local ReplicatedStorage = game:GetService("ReplicatedStorage")
local RemoteEvent = ReplicatedStorage:WaitForChild("ChangePlayerTextBoxText")
RemoteEvent.OnServerFired:Connect(function(player,Test) -- Remote Event must first value must be always to be player you can change it name how you want it just the player who fired event, next is think with which you fired event --
print(Test)
end)