Does the person's home country passport office have any way to check, possibly by scanning data transmitted to your passport by Schengen countries?
No. Countries do share immigration data, but they'll never do that
Whether by this SIS or not by SIS, how can someone check if they are banned from a Schengen country for an overstay after leaving the Schengen area?
GDPR in the EU grants you the right to access, rectify, and delete your personal data, and SIS is not an exception :
From the EU
What rights do people have regarding their data stored in SIS?
If a person believes that their personal information has been misused,
needs to be corrected or deleted, they may make use of the data
protection rights recognised in the SIS legislation. These are:
- the right to access personal data stored in SIS
- the right to correct inaccurate personal data or have unlawfully stored personal data be erased
- the right to bring proceedings before the courts or competent authorities to correct or erase personal data or to obtain
compensation for any damages resulting from breaches of data
The SIS legislation gives a person the right to initiate action before
the competent authority, including a court, under national rules to
obtain information, access, rectify, erase or obtain compensation (if
the person suffers damages) in connection with an alert relating to
him or her.
Countries using SIS have mutually agreed to enforce final decisions
handed down by the courts or authorities. This agreement means that a
decision taken by a court or competent authority in one country should
be recognised and enforced in all the countries that use SIS.
These rights can be exercised in any country that uses SIS. The
national procedures and contact points for access requests have been
compiled in a comprehensive guide
Follow the guide linked above
for an overstay
Overstays are calculated from the passport stamps only, if an overstay had occurred, the guard checking you on exit would inform you of the situation as you'll may be detained, served paperwork, have your visa cancelled... or any combination of all that