Currently, there is no option to allow/disallow message sending between users. This creates confusion on the expected behavior to enable/disable under the "Personal messages between users" option under Notification settings (Site Administration > Messaging > Notification Settings).
In the codes, there is a config variable "moodle_instantmessage_disabled" that seems to be responsible for two kinds of switches:
(1) whether message sending is disabled entirely
(2) whether message is sent and the notification is disabled (reminder feature will stay silent)
MDL-68070, a fix is proposed to allow messages be sent but notification to be suppressed. As related to this fix, we suggest to include a new toggle option to allow/disallow message sending between users, with the introduction of a new config variable (so as not to overload the use of "moodle_instantmessage_disabled"). The addition of the new toggle option should provide clarity to cover the two possible cases mentioned above.