Does a script in a player's backpack run on the client or the server?

It's neither actually.

To determine if a script runs on the client or on the server purely depends on the type of script that is being used.

LocalScripts will run on the client side; however, these scripts can only be ran in Backpack, PlayerGui, PlayerScripts, or inside the Player's character in order for them to work. These scripts can be determined in studio if they have a little person icon on top of the script in explorer. LocalScripts can not affect the server if FilteringEnabled is on, any changes you make to the server through a LocalScript become invisible to other players.

Scripts will run in Workspace, a Player's Backpack, or ServerScriptService ideally. These scripts just have a script icon on them in Studio. These scripts run for the Server and can change the server in FilteringEnabled environments.

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