Well, that's not quite constructive, is it? What to DO then? Simply put, have a dedicated user do all the Homebrew tasks, brew install, brew uninstall, brew update, etc., and others just use the software that's maintained by Homebrew.
This is actually a piece of advice that comes from this post. Not only did the author introduce her idea and elaborate her reason, but she also explained why other popular practices from Internet are not great ideas, which is really helpful to understand the matter thoroughly. Great post.
I used to share a laptop with my wife and tried sharing a Homebrew installation across both user accounts. Nasty business messing with permissions. I've recently got the chance to refresh that very laptop. I followed the advice and use my account solely to manage software with Homebrew. Her account just uses whatever software that Homebrew installs, Google Chrome, Microsoft Office, etc. It just feels right, and it's been working like a charm.