I’m not really sure. Here is my understanding:
rsync [options] src [dest]
So your command:
rsync -avz ~/live/public/system/ firstname.lastname@example.org:~/live/public/system/
Copies from the local ~/live/public/system/ directory to the ~mastodon/live/public/system/ directory on example.com
-a gives you “archive”, -v gives you “verbose” and -z gives you “compress”.
So that should leave you with the same file structure on example.com in the directory ~mastodon/live/public/system/ as you have on ~/live/public/system/ and you should get a list of each file as it transfers.
I hope that’s somewhat helpful.