Whether or not you yourself have to pay the Rundfunkbeitrag (broadcasting contribution) depends on whether or not the Beitragsservice (formerly GEZ) considers your place of residence to be its own flat or not. For the purpose of this answer, I am assuming that your landlord is law-abiding and pays the Rundfunkbeitrag for the flat he is inhabiting.
The main page of rundfunkbeitrag.de gives a first hint in a slightly hidden manner: one occurance of Wohnung (flat or place of residence) is highlighted as a link and clicking on it reveals the following pop-up text:
Eine Wohnung ist eine ortsfeste, baulich abgeschlossene Einheit, die zum Wohnen oder Schlafen geeignet ist oder genutzt wird, einen eigenen Eingang hat und nicht ausschließlich über eine andere Wohnung begehbar ist. Beitragsfrei sind Zimmer oder Wohnungen in Gemeinschaftsunterkünften wie Internaten oder Kasernen.
Loosely translated (I am not good at legalese):
A place of residence is a nonportable structually separated entity that can be used or is being used for living or sleeping, has its own entrance and can be entered by means other than exclusively via a different flat. Single rooms or flats in common housing such as boarding schools or barracks.
Clicking further reveals an English language PDF with information written explicitly for students but whose contents are also applicable to the general public as the only ‘special’ thing about students is that they may not have to pay if they receive the government-funded student aid BAFöG:
I live in a student hall of residence. What do I need to do?
The licence fee of € 17.50* per month is payable for rooms in student halls of residence that are accessed via a public corridor. Single Rooms count as a residence - regardless of whether they have their own bathroom or kitchen.
If you live in a room of a shared flat in a student hall of residence, only one person per shared flat needs to pay the licence fee. However, this is only the case if the shared flat is separated from a public corridor or starwell by a door and only those living in the flat have a key for the door.
Condensing this all into a final answer:
Say you have a single room or a single room with your own hallway and bathroom in one corner of your landlord’s flat. You can lock the door to your room(s) with a key only you should have. However, unless you climb through the window there is no way of getting in or out without traversing the flat of your landlord — whether it be the hallway only or whether you have to go through the living room.
This does not count as a Wohnung, you do not have to pay.
Say we are talking about a house with a front door and two doors leading away from the general hallway. One leads to your rooms, the other to the landlord’s rooms. While only the two parties have a key to the front door you can typically get in and out without them noticing.
You may get lucky but expect to have to pay for this as it would count as a Wohnung
Note that you will have to pay independently of whether your rooms have their own bathroom or not.
If the situation is as described in the first bullet point, you will need to ask your landlord for their Beitragsnummer (contribution number) and enter that into the field as requested. This number serves as confirmation that your landlord does indeed pay the contribution as is required.
