B737 NG First Officer reported returning to departure airport after experiencing a pressurization issue related to a cabin door not being fully closed.

Narrative

I was pilot flying and the Captain (CA) was the pilot monitoring (PM). On climb out the lead Flight Attendant (FA) at door 1L called flight deck and told the CA that door 1L was louder than normal. The noise was somewhat noticeable on the flight deck; but I was wearing noise cancelling headset so I would say it was only slightly louder than normal. The CA relayed the FA's information but did not suggest we return to ZZZ at this point or change our plan to go to ZZZ1. At some point between the FA's call; and before 10000 ft.; I stopped hand flying and turned on the AP. The airplane was light due to the less than 1 hour flight; so it was climbing rather quickly (approximately 3000-3500 fpm). It did not take much time after the FA's call that we passed through 10000 ft. MSL. As we climbed through approximately 12000 ft. the cabin altitude light came on with the associated horn. I asked the CA to request a descent to at least 10;000 feet and suggested we initiate the immediate action items and don our oxygen masks. We donned our oxygen masks and established communication using the overhead speakers.ATC gave us an initial descent to 10000 ft. At the time; I did note the highest cabin altitude we reached was 11000 ft. I used the AP B and LVL CHG with speed brakes to descend to 10000 ft. The CA requested a return to ZZZ with ATC and then gave me the radios so he could coordinate with the FAs and make a PA to the passengers. At some point after initiating the descent below 10000 ft. we both agreed that it must be that the main cabin door was not fully closed; despite the FWD ENTRY light being extinguished. ZZZ ATC gave us a further descent and vectors to the airport while the CA coordinated with the FAs. During this time I quickly set up for the visual approach backed up by the ILS. When the CA checked back in with me; I suggested we run the CABIN ALTITUDE WARNING checklist and he said he did not think it was necessary. He took back the radios and spoke to ZZZ regarding the return. Prior to our final descent; I suggested one more time that we run the CABIN ALTITUDE WARNING checklist and he again disagreed. I quickly briefed the visual approach and asked for a DESCENT checklist. We configured normally ran the LANDING checklist and made an uneventful landing. Due to an influx of traffic at that time; no gates were available when we talked with operations and they temporarily parked us on an apron.