Who is responsible for measuring the customer satisfaction in large organizations, is it the customer service department or the marketing department?

The answer is both, as the marketing department are responsible for the company image and will measure the customer satisfaction levels on-line, complaints box outside if there was a reception disk, sending the customers follow up emails to thank them for dealing with their business and ask them to perform a customer satisfaction survey and to help them improve better, and by monitoring their online presence and their social media channels to ensure the word of mouth is spreading and increase reach out or by any tools necessary according to the business nature, As for the Customer care department they are in charge of making sure the customer satisfaction rates are maintained high so the marketing department is monitoring its performance, but that does not mean that the customer care manager or supervisor does not follow up on customers with calls for loyal customers to check on the quality of service and how the company can improve their services next time, these kind of tactics tend to send a message about the company that they care for the best interest of the customers however you can overuse the tactics to come across as nagging or annoy the customer.
I hope I was able to help with my answer but its a broad subject you need to specify more in order to give better suggestios

the responsible entity for measuring the customers satisfaction level should be the one who has the direct contact and getting the first impression from the client which is the customer service ,then that index has to go to the marketing department to analyse it in order to keep on going or revise their campaigns .

my ans is both marketing is planing and customer service dept is executing so both are hand to hand together

Its is a responsibility of the entire company as a unit, Not One department.....If a customer contacts Marketing or Sales or Customer Service or even the company's phone operator looking for help, then the responsibility falls on that department and on that individual to take care of that issue/customer.
You absolutely can't bounce around a customer and ask them to call this number or that number or that individual.
Its you who need to get the information from the customer and then get in touch with the right department who can resolve that customer's issue in a timely fashion and ensure that the issue is resolved to the cusotmer's satisfaction.
By bouncing around the customer and pointing finger to another department, doesn't pose a good image of the company to that customer.
You as an individual need to take the responsibility and do the follow up.
To the customer, you are the company, you need to take the accountability and  Always DO THE RIGHT THING.
Always ask 'Is there anything else I can do for you', 'Have I answered all your questions', Are you satisfied with our customer service today'.
  Measurements can be done by sending the surveys to customer at anytime by any department but if all departments take care of the customer, the results will be far more positive. 
I can go further in detail if you can provide the environment/settings for which the Customer Service is needed.
I know its easier said then done but it is being done, examples, Costco Warehouse Club, Nordstrom, T-Mobile(to some extent) etc.
These companies don't know how to say NO to a customer and will bend over backwards to accomodate you even if it is not their departments issue.
They take care of their employees (they treat their employees as their internal customers) and their external customers.

the question is aimed to differentiate the responsibility of measuring the level of customer satisfaction.
We don't see students monitoring their own examination, nor professionals applying for certification conducting their own certification pass.
While the customer service busy serving the customers at their best, they will need third party to measure the satisfaction level and segment the need or gap through survey's directly with the customers.
Those findings then can be cross examined by the head of CS & Head of Marketing or quality control service head in order to find better ways improving the customer experience.
unfortunately, in newly developed markets such as the Middle East, the responsibility passed to the CS head of measuring his own performance in order to cut cost, this usually add more stress on the CS department and never produce authenticated findings.
in such of gap in CS performance, it does not mean that the CS department or the head are not performing, it could be over load per executive which affect the quality of service provided or lack of training due to cut on budget etc.
hence the measurement through third party are better for the correction of actions required, where management will be facing reality and they will have to increase the budget for additional resources or further training if required.

Both Marketing & Customer Care works hand in Hand.
Marketing Department brings in a potential Customer.
Sales converts the Potential Customer to a Customer and it is the Customer Care which needs to retain and delight them.
Even if one fails, everyone else fails.
This is a collective responsibility and not an individual responsibility.
In the case of Customer Satisfaction it is the entire Company's responsibility and not just one particular department.
The customer care department is a entry for the customer to get his needs / wants / service to be attended to.
Rest all would work in support of customer care to ensure that the customer is delighted and not just satisfied.

Customer satisfaction measuring is the Marketing dept.responsibility, the customer care maybe oneof the feedback sources

Marketing department sets the strategy and implements.

Customer service department monitors and checks the level of customer satisfaction.  

depends on company department hierarchy and individuals roles and responsibilities.

but measuring customers satisfaction comes under sales and marketing department for SURE

In organization there is no one best solution, so that deciding on work break down and the formation of functions depends on the unique case of the organization.
One of the features of excellent organizations is team work so I prefer to handle the customer satisfaction issue through a team formed for this purpose

Marketing department