Schedule K is a summary schedule of all shareholders' shares of the corporation's income, deductions, credits, etc. and is on page 3 of the 1120S. It is integral to the 1120S so it may not have been noticed that it is named Schedule K on that page.
For a copy please see http://www.irs.gov/pub/irs-pdf/f1120s.pdf
The amounts on Schedule K are divided into the distributive shares (for multiple owners) that are shown on the K-1. For a single owner the Schedule K and K-1 will have the same amounts, since 100% of the Sch. K amount is going to that one K-1.
From the instructions at http://www.irs.gov/instructions/i1120s/ch02.html#d0e2746
"Schedule K-1 shows each shareholder's separate share. Attach a copy of each Schedule K-1 to the Form 1120S filed with the IRS. Keep a copy for the corporation's records and give each shareholder a copy.
Give each shareholder a copy of the Shareholder's Instructions for Schedule K-1 (Form 1120S) or specific instructions for each item reported on the shareholder's Schedule K-1. "
So, yes, you do need the K-1 as part of the 1120S to be filed and in the corporate records.
No, you technically do not need to give the K-1 to the shareholder if the corporation gives specific instructions of what the shareholder must report. But, it makes a very good support document for the taxpayer's 1040 copy as it lists all of the items that are passing to the shareholder from the corporation.
Indeed, the Schedule K-1 is like a statement showing each shareholder's separate share of the items passed through from the corporation.
As always, please ask if you need more discussion or clarification.