When it comes to executing operations, the presence of conditions is unrelated to whether you choose callbacks or promises. Using plain callbacks for asynchronous operations is considered subpar, and leveraging promises whenever possible is recommended.

To enhance your token method with promises, you can implement the following approach:

app.factory('OauthService', function($http, $localStorage, $q) {
    return {
        token : function(callback){

            // Retrieving current token
            access_token = $localStorage.getObject('access_token');
            // Fetching current identity
            identity_token = $localStorage.getObject('identity_token');

            // If no user is logged in
            if(isObjectEmpty(identity_token)){

                // If access_token doesn't exist or is expiring soon
                if( isObjectEmpty(access_token) || 
                    Date.now() > (access_token.expires_at - (600*1000)) ){

                    // Generating an anonymous access_token using a promise
                    return $http
                        .get(domain+'/oauth/v2/token?client_id='+public_id + 
                             '&client_secret=' + secret + '&grant_type=client_credentials')
                        .then(function (response) {
                            $localStorage.setObject('access_token', {
                                key: response.data.access_token,
                                type: 'anonymous',
                                expires_at: Date.now() + 
                                       (response.data.expires_in * 1000)
                            });

                            return response.data.access_token;
                        });
                }
            }

            // If user is logged in
            else {
                // Processing when access_token is missing, about to expire, or is of anonymous type
                if( isObjectEmpty(access_token) || 
                    Date.now() > (access_token.expires_at - (600*1000)) || 
                    access_token.type == 'anonymous' ){

                    // Creating an access_token incorporating identity details through a promise
                    return $http
                        .get(domain+'/oauth/v2/token?client_id='+public_id+'&client_secret='+secret + 
                             '&api_key='+identity_token+'&grant_type=http://oauth2.dev/grants/api_key')
                        .then(function (response) {
                            $localStorage.setObject('access_token', {
                                key: response.data.access_token,
                                type: 'identity',
                                expires_at: Date.now()+
                                      (response.data.expires_in * 1000)
                            });

                            return response.data.access_token;    
                        });
                }

            }

            // Return existing access token (if previous token remains unchanged in type or hasn't expired)
            return $q.when(access_token.key);    
        }
    };
});

You can streamline the code further by refactoring it as shown below:

app.factory('OauthService', function($http, $localStorage, $q) {
    function expiresSoon(access_token) {
        return Date.now() > (access_token.expires_at - (600*1000));
    }

    function getAccessToken(url, type) {
        return $http
            .get(url)
            .then(function (response) {
                $localStorage.setObject('access_token', {
                    key: response.data.access_token,
                    type: type,
                    expires_at: Date.now() + 
                        (response.data.expires_in * 1000)
            });

                return response.data.access_token;
            });
    }

    return {
        token : function(callback){

            // Retrieve current token
            access_token = $localStorage.getObject('access_token');
            // Retrieve current identity
            identity_token = $localStorage.getObject('identity_token');

            // If no user is logged in
            if(isObjectEmpty(identity_token)){

                // If access_token doesn't exist or is expiring soon
                if( isObjectEmpty(access_token) || expiresSoon(access_token) ) {
                    var url = domain + '/oauth/v2/token?client_id=' + public_id + 
                              '&client_secret=' + secret + '&grant_type=client_credentials';
                    // Create an anonymous access_token using promises
                    return getAccessToken(url, 'anonymous');
                }
            }

            // If user is logged in
            else {
                // Processing when access_token is missing, about to expire, or is of anonymous type
                if( isObjectEmpty(access_token) || 
                    expiresSoon(access_token) || 
                    access_token.type == 'anonymous' ){

                    var url = domain+'/oauth/v2/token?client_id=' + public_id+
                              '&client_secret='+secret + 
                              '&api_key='+identity_token + 
                              '&grant_type=http://oauth2.dev/grants/api_key';

                    // Creating an access_token with identity using promises
                    return getAccessToken(url, 'identity');
                }
            }

            // Return existing access token (if previous token remains unchanged in type or hasn't expired)
            return $q.when(access_token.key);    
        }
    };
});