If you do decide to do a private sale, I prefer to list on community email listservs, Nextdoor (which does try to enforce residency, supposedly requires real names, and allows you to limit the area to which your for sale item is posted), and maybe post it on Farcebook, but NOT the Marketplace. Last time I posted a car on FB I made a regular post to friends and asked them to share. Those methods reduce the noise and improve the chances that you have a connection (maybe 3rd or 4th degree) to the person. When I got responses on community groups to people I didn't find a connection to, I could see their posting history and that they were involved in and knowledgeable about our area, which gave me confidence that they were who they said they were.
But then Carmax and Carvana are a lot less hassle for somewhat less money, so it really depends on what your priorities are. There's nothing wrong with paying more (or, rather, getting less) in exchange for speed, safety, predictability, and convenience.