You can't use $HOME
in the plist
but you can configure it easily enough.
Once the plist is loaded on the proper computer (lets assume it is at ~/Library/LaunchAgents/local.testing.plist
for the sake of discussion)
All you need to do is enter two commands at the command line
defaults write "$HOME/Library/LaunchAgents/local.testing.plist" \
StandardOutPath "$HOME/launchd.stdout.log"
defaults write "$HOME/Library/LaunchAgents/local.testing.plist" \
StandardErrorPath "$HOME/launchd.stderr.log"
That will add the two log definitions with the full path, specific and correct for each computer.
It will also replace any existing values, so you don't need to worry about duplication.
launchd
.