No, Slack does not notify other users when you save a message.
Saving a message in Slack is a private action that only you can see.
It allows you to bookmark important messages for easy access later, without alerting the person who sent the message or anyone else in the channel.
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To save a message, hover over the message you want to save, then click on the bookmark icon "Save for later".
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To access your saved messages, click on "Later" in the left navbar.