Local does a pretty good job of scrubbing private info from the logs and the errors it produces, however there’s always the possibility that something private can come through. Because these are public forums, always review the screenshots you are sharing to make sure there isn’t private info like passwords being displayed.
I noticed that in the Local logs, there is an error message “Error checking for latest version of WordPress” - which could mean that something on your machine is blocking Local’s request to WordPress.org. Do you possibly have a Firewall or Antivirus software that could be blocking that?
Finally, sometimes these errors occur with a theme or plugin generating PHP errors - are you experiencing this error when you try to install any plugin or a specific one?