I have manually added a search peer using the web manager > distributed search > search peers
The peer always has a status of Down. Replication status Initial
How do I troubleshoot this?
From the search page, the error message shows the correct name of the search peer so the two splunks must be talking to each other. It doesn't show an authentication error.
I wonder whether this is an issue with bundle replication. Try having a look at the questions in this post (i.e., are you seeing the same error message?).
You can look for bundles in the target machine's $SPLUNK_HOME/var/run/searchpeers to see if it's arriving there. There should be a .bundle file, as well as an unpacked directory with its contents.