It works in functions.php too, although I prefer using a custom plugin, to avoid loosing customizations when updating the theme.
You must not keep the default sharing label in the Jetpack plugin options because Jetpack makes a test against this default value and does not save it in the database. Also you must not leave it empty (as Polylang does not look for empty strings). Any other value should work.
I did not test in multisite installation.