Google-sheets – make a sheet private in a Google Sheets

Don't use several onEdit-functions. Places place both IF-statements in the same tag.

Also, you shouldn't have semicolons after the IF-statements brackets.

function onEdit() {
 var s = SpreadsheetApp.getActiveSheet();
 if( s.getName() == "General Meeting Attendance Roster" ) { //checks that we're on the correct sheet
       s.getRange('S6').setValue("Roster last updated: " + (new Date()));
  }

 if( s.getName() == "Master Chapter Roster" ) { //checks that we're on the correct sheet
       s.getRange('K1').setValue("Chapter Roster last updated: " + (new Date()));
  }
}

Yes, this is possible using an installable trigger, which runs as the user who created the trigger (not as the user at keyboard). Here is a simple example which compares the content of cell A1 in the current sheet with the content of cell A1 in the sheet "Sheet1" of some master spreadsheet. If they match, the background of A1 is set to red.

function compare(e) {
  var range = e.range;
  if (range.getA1Notation() == 'A1') {
    var masterSheet = SpreadsheetApp.openByUrl( your URL ).getSheetByName('Sheet1');
    if (range.getValue() == masterSheet.getRange('A1').getValue()) {
      range.setBackground('red');
    }
    else {
      range.setBackground('white');
    }
  }
}

After entering this script, add a trigger via Resources > Current Project's trigger. Under Events, select "From spreadsheet", "On edit".

As said above, this means that when triggered, the function compare will run as if it's run by you, and have the permissions that you have: in particular, the permission to access the master spreadsheet.