If I am reading your facts correctly, you can't just provide the shareholder's with the necessary state information. Depending on your state activity, if the activity rises to the point in which the S corp has "nexus", then you will be required to file a state return.
In addition to the potential required state filing at the S corp level, as @tagteam pointed out, the S corp may be able to file a composite return and remit the tax on behalf of the shareholder's. This would eliminate each shareholder from having to file a separate 1040 in each state (other than their state of residency).
In some cases the composite return may be part of the S corp state filing and in other cases it may be a completely separate return.
As you have most likely figured out, state filings vary dramatically and are not for the faint of heart. You may want to consider getting some professional input at some point in time.