Messages shouldn’t be displayed in the Comments section at all, this would be a bug. Yes, messages are implemented as hidden comments of “hp_message” type (to avoid creating a new database table in WordPress), but they shouldn’t be displayed as comments. They are displayed as comments when you disable HivePress Messages because the HivePress Messages extension stops restricting the message visibility so WordPress thinks that “hp_message” comments are regular ones.
Please try using monitoring instead, there may be a delay of about 1 hour in displaying other’s conversations but all the conversations should appear. If they are not please send steps for reproducing this issue or temporary WP access to support@hivepress.io and I’ll check this.