On Mastodon, you can browse other servers’ Local feeds by going to the server’s website and clicking the “Local” link. You don’t need to log in, anyone can look at the feed because Local feeds only show public posts anyway.
What about following another server’s Local timeline?
Some third party Mastodon apps (Fedilab and Subway Tooter on Android, Ice Cubes on iPhone/iPad) let you follow the Local timelines of other servers. The website interface and the official apps do not have this feature yet.
Many, many people have asked for this feature to be added to the website interface. If you want to vote for this and you are comfortable using Github, click here and give a thumbs up to the first post in this thread ⧉. This will let the developers know that the feature is wanted.