I don't think there's any way to tell if something is private other than doing research on it and asking around. I think privates in the new CW shops out of the CS have a lock on them (and you can only buy them with a password given to you by the shop owner), but, otherwise, I don't think there's another way. Privates are really a user-run thing, not a site-run thing, so the site doesn't really have a system to keep track of privates vs publics.
As far as I know, unless you know 1) who the artist is, 2) the link to their CW thread, if they have one, and 3) if the item is still listed in their thread and -says- private, there's no way to know. "Private" is a user-made thing, not a site thing.
Edit: Beat to it while I was verifying :P
Thank you both! Looks like I have quite a few to do some research on, lol
Happy to help!
I think the best thing for you to do would be making a thread in the Item Pricing forum and listing the items you'd like to know about there. Just ask for what they generally go for and mention that you'd like to know if any are private. I've seen that forum help out a lot of people. :)