✓ Accepted answer
Anything inside a LocalScript is ONLY run on the client. This means that the changes will not replicate to every other player in the game.
Since you are adding 1 point every second to the player through a LocalScript, you aren't actually giving them any points.
What you need to do is set-up a leaderboards script in ServerScriptService and run your loop on that value instead.
To show the user how many points they have on a GUI, you will want to put a LocalScript inside the TextLabel and get it to update the Text whenever the points increase.
local Player = game:GetService("Players").LocalPlayer
local points = Player.leaderstats.Points
local PointsText = script.Parent
Points:GetPropertyChangedSignal("Value"):Connect(function()
PointsText.Text = points.Value
end)