The AirTag relies on Bluetooth to send its location information to the Apple servers. In order to get there it would first send that signal to a nearby iPhone that is participating in the "Find My Network," and has either a cellular or wireless connection to the Internet. That network is enabled by default: Settings > Apple ID > Find My iPhone > Find My Network
That iPhone would need to be within 30-40' of the AirTag to receive that signal. This is where your neighbor is getting that notification. However, as Chattanoogan has provided you, this should only happen if you (your iPhone actually) is separated from the AirTag that it is paired with after a specific amount of time/distance.
One option, that Apple provides you, is to disable getting notifications for locations you specify. For example, if you have an AirTag that you leave at home, and you go somewhere else with your iPhone, you can identify that your home is a location where your AirTag will not notify you.
To enable this: Find My > Select the Items tab > Select your AirTag > Notify When Left Behind (enable). Then under "Notify Me, Except At," select New Location, and then, select the location on the map provided.