Injecting $scope into an angular service function()

The $scope that you see being injected into controllers is not some service (like the rest of the injectable stuff), but is a Scope object. Many scope objects can be created (usually prototypically inheriting from a parent scope). The root of all scopes is the $rootScope and you can create a new child-scope using the $new() method of any scope (including the $rootScope).

The purpose of a Scope is to "glue together" the presentation and the business logic of your app. It does not make much sense to pass a $scope into a service.

Services are singleton objects used (among other things) to share data (e.g. among several controllers) and generally encapsulate reusable pieces of code (since they can be injected and offer their "services" in any part of your app that needs them: controllers, directives, filters, other services etc).

I am sure, various approaches would work for you. One is this:
Since the StudentService is in charge of dealing with student data, you can have the StudentService keep an array of students and let it "share" it with whoever might be interested (e.g. your $scope). This makes even more sense, if there are other views/controllers/filters/services that need to have access to that info (if there aren't any right now, don't be surprised if they start popping up soon).
Every time a new student is added (using the service's save() method), the service's own array of students will be updated and every other object sharing that array will get automatically updated as well.

Based on the approach described above, your code could look like this:

angular.
  module('cfd', []).

  factory('StudentService', ['$http', '$q', function ($http, $q) {
    var path = 'data/people/students.json';
    var students = [];

    // In the real app, instead of just updating the students array
    // (which will be probably already done from the controller)
    // this method should send the student data to the server and
    // wait for a response.
    // This method returns a promise to emulate what would happen 
    // when actually communicating with the server.
    var save = function (student) {
      if (student.id === null) {
        students.push(student);
      } else {
        for (var i = 0; i < students.length; i++) {
          if (students[i].id === student.id) {
            students[i] = student;
            break;
          }
        }
      }

      return $q.resolve(student);
    };

    // Populate the students array with students from the server.
    $http.get(path).then(function (response) {
      response.data.forEach(function (student) {
        students.push(student);
      });
    });

    return {
      students: students,
      save: save
    };     
  }]).

  controller('someCtrl', ['$scope', 'StudentService', 
    function ($scope, StudentService) {
      $scope.students = StudentService.students;
      $scope.saveStudent = function (student) {
        // Do some $scope-specific stuff...

        // Do the actual saving using the StudentService.
        // Once the operation is completed, the $scope's `students`
        // array will be automatically updated, since it references
        // the StudentService's `students` array.
        StudentService.save(student).then(function () {
          // Do some more $scope-specific stuff, 
          // e.g. show a notification.
        }, function (err) {
          // Handle the error.
        });
      };
    }
]);

_{One thing you should be careful about when using this approach is to never re-assign the service's array, because then any other components (e.g. scopes) will be still referencing the original array and your app will break.
E.g. to clear the array in StudentService:}

/* DON'T DO THAT   */  
var clear = function () { students = []; }

/* DO THIS INSTEAD */  
var clear = function () { students.splice(0, students.length); }

See, also, this short demo.

LITTLE UPDATE:

A few words to avoid the confusion that may arise while talking about using a service, but not creating it with the service() function.

Quoting the docs on $provide:

An Angular service is a singleton object created by a service factory. These service factories are functions which, in turn, are created by a service provider. The service providers are constructor functions. When instantiated they must contain a property called $get, which holds the service factory function.
[...]
...the $provide service has additional helper methods to register services without specifying a provider:

• provider(provider) - registers a service provider with the $injector
• constant(obj) - registers a value/object that can be accessed by providers and services.
• value(obj) - registers a value/object that can only be accessed by services, not providers.
• factory(fn) - registers a service factory function, fn, that will be wrapped in a service provider object, whose $get property will contain the given factory function.
• service(class) - registers a constructor function, class that will be wrapped in a service provider object, whose $get property will instantiate a new object using the given constructor function.

Basically, what it says is that every Angular service is registered using $provide.provider(), but there are "shortcut" methods for simpler services (two of which are service() and factory()).
It all "boils down" to a service, so it doesn't make much difference which method you use (as long as the requirements for your service can be covered by that method).

BTW, provider vs service vs factory is one of the most confusing concepts for Angular new-comers, but fortunately there are plenty of resources (here on SO) to make things easier. (Just search around.)

(I hope that clears it up - let me know if it doesn't.)

Instead of trying to modify the $scope within the service, you can implement a $watch within your controller to watch a property on your service for changes and then update a property on the $scope. Here is an example you might try in a controller:

angular.module('cfd')
    .controller('MyController', ['$scope', 'StudentService', function ($scope, StudentService) {

        $scope.students = null;

        (function () {
            $scope.$watch(function () {
                return StudentService.students;
            }, function (newVal, oldVal) {
                if ( newValue !== oldValue ) {
                    $scope.students = newVal;
                }
            });
        }());
    }]);

One thing to note is that within your service, in order for the students property to be visible, it needs to be on the Service object or this like so:

this.students = $http.get(path).then(function (resp) {
  return resp.data;
});

Well (a long one) ... if you insist to have $scope access inside a service, you can:

Create a getter/setter service

ngapp.factory('Scopes', function (){
  var mem = {};
  return {
    store: function (key, value) { mem[key] = value; },
    get: function (key) { return mem[key]; }
  };
});

Inject it and store the controller scope in it

ngapp.controller('myCtrl', ['$scope', 'Scopes', function($scope, Scopes) {
  Scopes.store('myCtrl', $scope);
}]);

Now, get the scope inside another service

ngapp.factory('getRoute', ['Scopes', '$http', function(Scopes, $http){
  // there you are
  var $scope = Scopes.get('myCtrl');
}]);