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But it doesn't do this right away.I was traveling with a group of people that had AirTags, and I only start getting alerts by day three.

You shouldn't get them ever if you are traveling with the owner of the AirTag. The criteria for notification is:

1. the airtag is following you

2. the owner of the airtag does is not near the airtag

I get them within 30 to 45 minutes, consistently. You won't get them if the people are traveling with you.