No, that's not what happens. If your device detects the same AirTag traveling with you, despite not being paired to your account you will be notified about it,

No, that's not what happens. If your device detects the same AirTag traveling with you, despite not being paired to your account you will be notified about it, no matter what state it is in. /if/ the device is in lost mode you're provided with contact details.