Who gets to name them?

> The OpenStreetMap community operates under the “on the ground” principle, recording what is actually currently used in a particular area, giving pre­eminence to data collected in­ situ.

https://osmfoundation.org/w/images/d/d8/DisputedTerritoriesI...

Private entities get to use whatever name they want. In this case, there is the option to use the longstanding name that matches the one used by the other neighboring territory. Or to have a wait period after the name is changed unilaterally by executive order, because it'd be silly to change it back and forth. Or to put "a.k.a..." after it. Or to wait for the relevant government to update its own map and geographic names database first (https://edits.nationalmap.gov/apps/gaz-domestic/public/searc... is not yet updated as of now). Or to wait until local signage can be updated so it mostly matches the map (though I don't think Rochester will be in a rush to do this).

Well Derry/Londonderry is officially called Londonderry, but Irish republicans didn't like the London link so call(ed) it Derry. So during the troubles it was a thing to refer to is as Derry/Londonderry.