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On macOS, with bash installed from homebrew, I noticed that setting LC_MESSAGES seems to have some effect on the current shell's locale settings, but messages doesn't actually change until LC_MESSAGES is exported:

Unsetting LANG and LC_MESSAGES, I get an English error message as expected:

bash-4.4$ unset LANG LC_MESSAGES

bash-4.4$ if :;  fi

bash: syntax error near unexpected token `fi'

Setting LC_MESSAGES to an incorrect value gives an error about setlocale:

bash-4.4$ LC_MESSAGES=foo

bash: warning: setlocale: LC_MESSAGES: cannot change locale (foo): No such file or directory

So something changes when I set LC_MESSAGES. But setting it to a reasonable value has no effect:

bash-4.4$ LC_MESSAGES=ja_JP.UTF-8

bash-4.4$ if :;  fi

bash: syntax error near unexpected token `fi'

Until I export it:

bash-4.4$ export LC_MESSAGES

bash-4.4$ if :;  fi

bash: 予期しないトークン `fi' 周辺に構文エラーがあります

(All of this goes for LANG as well, it seems.)

The Bash manual's section on Bash Variables does not say LC_MESSAGES or LANG has to be exported (and most other variables listed there don't have to be exported).

Why is this?

Take a look at this answer for an explaination of the difference between a shell variable and an environment variable. In essence:

Setting a shell variable:

LANG=en_US.UTF-8

Setting an environment variable:

export LANG=en_US.UTF-8

You want to set environment variables for locale, as shell variables are private to the shell and won’t be passed to child processes.