How can I logout of the private store?
If you're using the password protection that comes with WooCommerce Private Store then there is no option to log out of the store. Instead, a cookie is set in the user's browser when they enter the password to unlock the store. The cookie lets them continue to access the store until it expires. Once the cookie expires, the user is 'logged out' of the store and has to re-enter their password to access it again.
The 'Password Expires After' option lets you change the number of days before the cookie expires. Set it to the minimum number of days if you want to 'logout' users as quickly as possible.
If you want people to be able to log into your store using their own individual user accounts, then you can use the Logged In Users option that comes with WooCommerce Private Store. This is an alternative to password protection. If you use this option, then you can let people log out of their user account. However, a cookie will still have been set and they will continue being able to see the WooCommerce content until it expires after the number of days specified on the settings page.