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 bugPixelmon 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.