Shoutbox

As matty said, scripts are always started globally in Messenger.
What account you use to sign in into Messenger doesn't matter.
In fact, scripts start to run even before you have signed in, and they continue to run after you manually signed out. They only stop running when Messenger is also stopped.

If you disable a script in one account, it will also be disabled for all other acounts you might use to sign in into Messenger.

There is nothing you can do (in a decent way) to just have a script loaded and running for only a specific account.

However, and Matty said this already, you can make or edit your script in such a way that it will only perform certain actions when a specific account is used to sign in. For other accounts it would still be loaded and running though, but it wouldn't perform its actions.

It highly and completely depends on the specific script how or when you would check for the signed in account. As such, we can not give you a solution for your problem, unless you tell us about what specific script you're talking about or give us the complete code of the script you've created.
But it in a nutshell, it always comes down to checking the signed in account (Messenger.MyEmail) before performing the intended action.

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Running multiple instances of Messenger doesn't have any influence on this. The principle stays the same: in the script you need to check the signed in user before performing the action.