I read on the pricing page that the files in the Free package are not private.
Is this true?
No it’s not true. All your files are private unless you share them.
Private projects is a completely different feature.
The private project is crossed out. 
Private projects are a completely different feature, unrelated to private files. Projects are not files. Don’t worry about it. You won’t need this feature. https://help.figma.com/hc/en-us/articles/360038006494-Create-a-new-project
Maybe I can check somewhere in the menu if someone has access to my file?
As in OneDrive, a small icon indicates that the file is shared.
Yes, you can check in the Share menu. Set file settings to “only people invited” to make it not accessible via link.
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Thank you for your help