The town building menu is inaccessible without the original town building owner when playing in local COOP. The original town owner must also be in town in order for the player 2 and beyond to access the build menu.
What this also means is if you delete all characters, including the original town owner, the build menu becomes completely inaccessible to the player, and they have to start a new save file in order to get access to the town building.
Steps to Reproduce
This is a demonstration of the local COOP player not having access to the town building:
Spoiler
This is what happens when you delete the OG town builder (I checked save slot 1 after this recording as well to confirm it doesn't work):
Spoiler
There are no comments to display.