Currently fish_greeting function is defined as:

	function fish_greeting
	if not set -q fish_greeting
	set -l line1 (_ 'Welcome to fish, the friendly interactive shell')
	set -l line2 \n(printf (_ 'Type %shelp%s for instructions on how to use fish') (set_color green) (set_color normal))
	set -g fish_greeting "$line1$line2"
	end

	if set -q fish_private_mode
	set -l line (_ "fish is running in private mode, history will not be persisted.")
	set -g fish_greeting $fish_greeting.\n$line
	end

	# The greeting used to be skipped when fish_greeting was empty (not just undefined)
	# Keep it that way to not print superfluous newlines on old configuration
	test -n "$fish_greeting"
	and echo $fish_greeting
	end

When $fish_greeting is empty, the $fish_greeting.\n$line will expand into a empty string, so fish --private won't show any message either.

I wonder if it makes sense to still show the private mode message ("fish is running in private mode, history will not be persisted."), even if $fish_greeting is empty.

Also, there's a slight inconsistency:

$ set -U fish_greeting
$ fish -P
(no message)
---
$ set -U fish_greeting ""
$ fish -P
.
fish is running in private mode, history will not be persisted.

For reference, there is a similar issue: #6299