Today the same thing happened to me again, and I was able to figure out the root cause.
For some reason there was an additional Subscriber role assigned to the Admin.
Taking it out, and it was working again.
I had another look, and the error message is gone, the subscriber role can now send messages.
Not sure why it works now.
Perhaps it had to do with another plugin that was causing conflicts with some other plugins, too, and for which I just installed an updated version.
Anyway, the issue is gone.