The idea would be to try to reduce the user response time by giving a badge to individuals or organizations that are very responsive to private messages. This way we would try to make sure everyone is more active and paying attention to the app.

The badge would be something like facebook has or had like "Very responsive to messages" badge that would appear on the profile or also below the posts so other users could see it.

Maybe this is a question to involve marketing as well to discuss it, since i dont fully understand the drawbacks of orgs or individuals feeling pressured by being more available here (Is this a concern?)

What i saw on a page i found but regarding fb, is that this features provides visitors an “expectation of the amount of time it takes your Page (in our case a user/owner of a post) to respond to messages” .

A kind of gamification also.