Unfortunately, to the best of my knowledge, it is not possible to send messages through the ArcGIS Online platform to other members. There is a notification system (i.e. you're notified when you have been added to a group), but not a messaging system.
You can use the ArcGIS API for Python to member information for people in your organization (see this blog post by Egge-Jan Pollé). However, I don't think this will work for people outside of your organization.
I found this idea about adding a messaging capability to ArcGIS Online. It's a pretty old post, but maybe it can be revitalized with some comments and up-votes. Or if this doesn't cover exactly what you want, I recommend making a new one! If you do, share it here so we can spread the word
Thank you Katherine for the encompassing inquiry. This issue (#communication) is pretty essential for professional collaboration. Is nobody else demanding that? Do you know if this #idea is in the pipeline?
Just based on the updates that Esri puts out, I haven't heard anything about this planned on being implemented. Again, I recommend either commenting on that idea I linked to and up-voting it, or starting a new idea to describe exactly what you are looking/hoping for. Esri really does look at those, so it's possible it could make a difference. (I actually got a call once from an Esri rep to follow up on a comment I made on an idea, because they were gathering user feedback to aid in their product plan)