jquery datatables selected row's data getting reset on paginated table while navigating on pages

Please take a look at my solution at JSFiddle.

HTML

<table id="test" class="display">
    <thead><tr><th>select</th><th>operation</th></tr></thead>
    <tbody></tbody>
</table>

<p>
    <input id="test-data-json" name="test_data_json" type="hidden">
    <button id="btn-submit">Submit</button>
</p>

Javascript:

//ajax emulation
$.mockjax({
   url: '/test/0',
   responseTime: 200,
   responseText: {
      "aaData" : [
         [{"id":1}, {"chk":"on"}, {"operation":["Modify", "Delete"]}],
         [{"id":2}, {"chk":"on"}, {"operation":["Modify", "Delete"]}],
         [{"id":3}, {"chk":"on"}, {"operation":["Modify", "Delete"]}],
         [{"id":4}, {"chk":"on"}, {"operation":["Modify", "Delete"]}],
         [{"id":5}, {"chk":"on"}, {"operation":["Modify", "Delete"]}],
         [{"id":6}, {"chk":"on"}, {"operation":["Modify", "Delete"]}],
         [{"id":7}, {"chk":"on"}, {"operation":["Modify", "Delete"]}]
      ]
   }
});

// Global variable holding current state of the controls in the data table
var g_data = {};

var $table = $('#test');
$table.dataTable( {
   "lengthMenu": [[2,25,50 ,100, -1], [2,25,50, 100, "All"]],
   "pagingType": "full_numbers" ,
   "paging":   true,
   "ordering" : true,
   "scrollY":false,
   "autoWidth": false,
   "serverSide": false,
   "processing": false,
   "info":     true ,
   "deferRender": true,
   "processing": true,
   "columns": [
      ["data", 1 ],
      ["data", 2 ]
   ],
   "ajax" : {
      "url": "/test/0",
      "dataSrc" : function(json){
         var data = json.aaData;
         for (var i = 0; i < data.length; i++){
            var chk_name  = 'chk_' + data[i][0].id;

            // If checkbox was never checked
            if(typeof g_data[chk_name] === 'undefined'){
               // Retrieve checkbox state from received data
               g_data[chk_name] = (data[i][1].chk === 'on') ? true : false;
            }
         }

         return data;
      }
   },
   "createdRow" : function( row, data, index ){
      var chk_name  = 'chk_' + data[0].id;
      var chk_checked = (g_data[chk_name]) ? " checked" : "";

      $('td:eq(0)', row)
         .html('<input name="' + chk_name +'" type="checkbox" value="1"' + chk_checked + '>');

      var select_name = 'select_' + data[0].id;
      html =
         '<select name="' + select_name +'">'
          + '<option value="">Select one</option>'
          + '<option'
          + ((typeof g_data[select_name] !== 'undefined' && g_data[select_name] === data[2].operation[0]) ? ' selected' : '')
          + '>' + data[2].operation[0] + '</option>'
          + '<option'
          + ((typeof g_data[select_name] !== 'undefined' && g_data[select_name] === data[2].operation[1]) ? ' selected' : '')
          + '>' + data[2].operation[1] + '</option>';
          + '</select>';

      $('td:eq(1)', row).html(html);
   },
});

$('#test tbody').on('click', 'input[type=checkbox]', function (e){
   g_data[this.name] = this.checked;
});

$('#test tbody').on('change', 'select', function (e){
   if(this.selectedIndex != -1){
      var value = this.options[this.selectedIndex].value;
      g_data[this.name] = value;
   }
});

$('#btn-submit').on('click', function(){
   $('#test-data-json').val(JSON.stringify(g_data));
   console.log($('#test-data-json').val());
});

I have slightly updated your code since it was a mix of new and legacy options. However I haven't edited legacy server response using aaData property, so you don't have to change your server-side script.

Basically, the solution is to use a variable (g_data in my example) to store/retrieve state of dynamic form controls.

Upon form submission, the data is stored in hidden INPUT element as JSON string.

Optionally, if form validation is needed, inspect the state of controls stored in g_data variable.