Those admin notices are only shown after plugin activation. The plugin runs two background tasks in order to update (if necessary) post and authorship counters. If you have WP Cron disabled or blocked in some way, the jobs won’t run and counters won’t be updated.
If those notices keep displaying even when the job is done, you may have a cache plugin. Try to flush admin cache. Does this make those notices to go away?
However, when adding a new post (or editing an existing one), metadata is set while saving the post. No cron job is run then. So post counters should be updated, even if they are not displaying a correct number. Let’s say John Doe’s post count is 8. But our counter display 3. If you add John to a new post, our counter should display 4. The count is not correct, but it’s been updated with last change. Is it working like this for you?
I mean, updating all post counters so they reflect the actual post count for each author is done as a background task. By default it is run on plugin activation. And you can also run it manually from: Authors > Tools > Authorship. But incrementing/decrementing post counters when editing a post, is done automatically on post save.
Please let us know so we can assist.