Currently there only exist one error notification in HS and that appears to be on the Salesforce Integration.
We had a lead syncing issue today on an account and apparently it had been not working for some time, with LinkedIn syncing, however no one in the HS Admin circle received a notification that there was an issue.
The business impact of this issue results in leads not being tracked properly and under estimating performance and pipeline. Had we not done a manual exercise of cross checking leads, we may have never known there was a problem.