It depends on what kind of freelancer you are. If you don't actually have in-demand skills, then the only invitations you will receive as a newbie are scams. So setting your profile to public won't help you.
Feelancers with in-demand skills or extensive Upwork experience can receive very lucrative invitations. That is why my profile is public, when I have time to accept new clients. With a public profile, I receive more invitations because I'm easier to find.
If you have in-demand skills, then there really is not a compelling reason to set your profile to private unless you're trying to hide from people.
And even then, it doesn't make sense. If you want to hide, then having a profile on Upwork is NOT a good idea. Because this is a VERY public website. A VERY large website. There are no privacy settings on Upwork that will guarantee your privacy.
The most that you can expect from the privacy settings is to control whether or not you appear in Upwork's search results, and whether or not users can view you profile all the time or only when logged in.
So privacy settings CAN control the amount of exposure you receive. But setting you profile to private can't guarantee nobody will learn you're on Upwork.
I decided to just make my profile available to Upwork users who are logged in, to avoid being scammed in the public setting. When I build a network of clients then I'll use the public setting. I'm fine with gaining slow momentum. I'm not interested in using the private setting, at least not at this stage. Middle of the road is fine right now. Thanks.