Hi,
Just want to share my experience with a locator, this might help you or others searching for solutions.
ESRI UK provide free locators specific for accessing UK locations, for example we use the British National Grid which they have provided freely as a locator.
I added this locator to our organisation via the AGOL settings > Utility services > Geocoding. Everything worked as expected.
Until...
A colleague created a web map which he wished to share publicly to a client. The map and data were fine but we noticed the search box showed none of the locators.
Turns out after much faffing we had to turn on Security > Policies > Allow anonymous access.
This allowed the search box to populate with locators on public maps. It also opens up our organisational home page to the public to show only the publicly shared layers/apps.