people i am with get notifications that an airtag is traveling with them and may be tracking them. Why would it do that if my airtag is registered to my phone? Is it a faulty airtag?

You are correct that your friends normally shouldn’t be alerted IF your paired iPhone (with network connectivity) is nearby.

So … does the “offending” Airtag show-up on your phone’s FindMy -> Items; is your phone’s Bluetooth “on”; are location services enabled; and do you have good mobile network coverage when this occurs?

Note too that this might also be caused if your friends don’t have good mobile network connectivity.

See also “What you need” here:

Set up your AirTag using your iPhone, iPad, or iPod touch - Apple Support

That's correct and the way it works. The AirTag is not paired with their iPhones or is not shared with your them so they will get an alert if one is following them. I believe you have to enable Family Sharing for an AirTag to be shared on multiple devices before IOS 17. You'll also most likely have to add iCloud+ for more storage. Otherwise, others that are near your tag that you don't include in your Family Sharing will get the alert that they're being followed, and they can only pause it for 24 hours. After IOS 17 you can share the AirTags with up to 5 more devices. iPhone 15 or newer is required for precise location.

You’re welcome.

I’m confident that will solve the issue.

Without bluetooth “on”, your phone isn’t capable of “reporting” your tag’s location to the FindMy network; so the network “assumes” that your device is NOT nearby.