Here is the issue report I opened on FE's issue tracker: https://github.com/ForgeEssentials/ForgeEssentials/issues/2583. That issue report contains my mod list, server latest.log, and server debug.log

Describe the bug
Pixelmon Broadcasts uses a permission for broadcasted chat messages, such as "pixelmonbroadcasts.notify" (link), that are not recognized by Forge Essentials. The only node that generates in Essentials' PermissionsList.txt is "command.pixelmonbroadcasts", this is an unrelated node. OPs receive the broadcasts, anyone else does not.

Response from spacebuilder2020
This is probably cause Pixelmon Broadcasts is not registering permissions properly in Forge. As long as a permission has been registered, it should show up in FE. (This is specific to Boolean permissions only. Forge Permission api does not support non Boolean permissions). I recommend making a bug report on the Pixelmon side and referencing this issue. In the case of a non Boolean permission, the mod will need to check FE's extended permission handler.

I'll have a look at this one when I start work on the next update. I'm not entirely sure why FE would behave differently to something like LuckPerms here, but some testing should hopefully tease out the problem or find an usable workaround.

Keep an eye on the ticket status! (just keep in mind that the next update may be a while out still)

With the new, re-written versions of Pixelmon Broadcasts, this should now be fixed. The latest version of 1.1.1. Please do let me know if you have issues with the permissions.