I'm not aware of existing libraries with such support, but one could use the standard JSON.parse method and then manually walk the result restoring the circular references - it'd just be a simple store/lookup based on the $id property. (A similar approach can be used for reversing the process.)
Here is some sample code that uses such an approach. This code assumes the JSON has already been parsed to the relevant JS object graph - it also modifies the supplied data. YMMV.
function restoreJsonNetCR(g) {
  var ids = {};
  function relink (s) {
    // we care naught about primitives
    if (s === null || typeof s !== "object") { return s; }
    var id = s['$id'];
    delete s['$id'];
    // either return previously known object, or
    // remember this object linking for later
    if (ids[id]) {
      return ids[id];
    }
    ids[id] = s;
    // then, recursively for each key/index, relink the sub-graph
    if (s.hasOwnProperty('length')) {
      // array or array-like; a different guard may be more appropriate
      for (var i = 0; i < s.length; i++) {
        s[i] = relink(s[i]);
      }
    } else {
      // other objects
      for (var p in s) {
        if (s.hasOwnProperty(p)) {
          s[p] = relink(s[p]);
        }
      }
    }
    return s;
  }
  return relink(g);
}
And the usage
var d = {
    "$id": "1",
    "AppViewColumns": [
        {
            "$id": "2",
            "AppView": {"$id":"1"},
            "ColumnID": 1,
        }
    ]
};
d = restoreJsonNetCR(d);
// the following works well in Chrome, YMMV in other developer tools
console.log(d);
DrSammyD created an underscore plugin variant with round-trip support.