Firstly, with MacOS Catalina, the basic ways to change the ownership of /usr/local are no longer allowed. For example:
$ sudo chown -R "$USER":wheel /usr/local
Password:
chown: /usr/local: Operation not permitted
$ sudo chown -R "$USER" /usr/local
chown: /usr/local: Operation not permitted
$ sudo chown -R $(whoami) /usr/local
chown: /usr/local: Operation not permitted
Hence, the popular answers above cannot be used. Secondly, however, taking a step back, if the main concern is to install or upgrade Homebrew, rather than wanting to change the permissions for /usr/local per se, then it may be overkill (like taking a sledgehammer to hammer a nail) to change the permissions for /usr/local. It affects your whole machine and other software may also be using /usr/local. For example, I have files related to maven and mySQL in /usr/local.
A more precise solution is to follow the instructions to install Homebrew, given at the Homebrew GitHub site, namely
/bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install.sh)"
which installs Homebrew inside /usr/local without changing ownership of /usr/local itself. Instead, Cellar, Caskroom, Frameworks, Homebrew, etc. are installed inside /usr/local. This seems to be a more elegant, precise solution in my opinion.