https://support.apple.com/en-us/HT212227
Idk, I don't use apple products. According to Apple: "If an AirTag that's separated from its owner is seen moving with you over time, or if your iPhone is awake and an AirTag that isn't with its owner for a period of time emits a sound to indicate that it has moved, this alert is displayed: AirTag Found Moving With You."
It seems there is a setting for having your phone "awake" that is required, I'm sure the prudent thief would have it enabled. It's possible that the Tag is synched to both of your devices so it wouldn't alert to that
Who knows with all of the new tech these days. Just something to think about and research before use.