It is a privacy violation to indicate the PM exists to an anon user, because anon users should not know there is content there.
Indeed there is:
Ooops! That page doesn’t exist or is private
You could edit this copy in Admin, Customize to make it more suggestive of login if you like.
I believe we discussed a site setting that would relax this requirement for sites that don’t care about the privacy implications, pretty sure there are existing discussions on this.