problems with user rights

I turned SEF off and caused the error. URL is "index.php?option=com_comprofiler&view=saveuseredit&Itemid=397"
When I click the menu item URL is "index.php?option=com_comprofiler&view=userdetails&Itemid=347" Access to "index.php?option=com_comprofiler&view=userdetails&Itemid=347" is public at the moment. Usually it is not public, it is registered.

What means: "match it to its Joomla menu item then adjust the access for that menu item."

What means: "match it to its Joomla menu item then adjust the access for that menu item."

Navigate to Menus > All Menu Items then find the menu item with an ID (far right column) of 397 as that's the Itemid used in the save profile edit URL. Ensure that menu item has proper permissions.

Menu item with ID 397 is published and public, but I still have the error

Don't know what more to suggest. You've a permissions problem somewhere most likely then. Set the user to Joomla standard Registered usergroup and see if that works fine. For your usergroup with elevated permissions ensure it's a child of Registered. Example as follows.

Public
- Registered
- - Custom

It doesn't work. I use an other user type and it works.

I use an other user type and it works.

It doesn't work with Registered or it doesn't work with your custom usergroup? All default permissions are setup for Registered already so should work fine. Can't really help you any further as it's a permissions issue you've likely created by adding custom usergroups outside of existing parent usergroups preventing inheritance from working so you have to completely setup all their permissions (that's just how Joomla permissions work and isn't specific to CB).