My PermissionsList.txt is not populated with new permissions from mods, although the modlist.txt file is being populated, just like the PermissionList_Items.txt and the PermissionList_Blocks.txt. I'm using the forge server for direwolf20 1.10 modpack (minecraft version: 1.10.2).

My current PermissionsList.txt
My current modlist.txt (there are much more mods, really)
My current REAL list of mods

I've tried to add a few rules for known permissions (like, for example, the com.auth.mod.commands.RegisterCommand permission for AuthMod, but it does not work at all - the console reports that I lack a permission to run this command).

Well, looks like it was a conflict with FTBUtilities. After removing FTBUtilities, everything went back to normal. The problem is - how can I achieve normal functionality with both mods active? I really don't want to stop using the chunk claiming system FTBU provides, the team system it uses, etc. Do I have to address this issue to the FTBU devs?

make sure you are on the latest version of ftb utils. There was a conflict with forge essentials but I think it was recently fixed.

Also update to the latest version of forge essentials
Make sure to follow the procedure documented here. Failure to follow this procedure will result in your report being closed.

There were some useful config setings FTBU devs send me that helped the situation. For anyone wondering, those were:

"ranks": {
    "override_chat": false,
    "override_commands": false,
    "enabled": false
  }

at the config.json file (located at /local/ftbu/config.json)

ok. Great