C-172 pilot reported experiencing a rough running engine while enroute and completed a precautionary landing at a private airport.

Narrative

I was PIC of Aircraft X. On the day of the [event]; our plan was to fly to ZZZ1. We were flying southbound along the shoreline and all parameters of the aircraft and engine were normal. A couple of miles northwest of ZZZ the engine started to make rough sounds affecting the performance of the aircraft. Immediately I reported to ATC that I needed to divert to the nearest alternate (ZZZ) due to engine roughness and clarified that I wasn't declaring an emergency at that time. I decided to descend and join a left base for Runway XX and contact ZZZ. While I was turning to join final; I contacted ZZZ and explained the situation regarding my flight. The ATC told me that I wasn't cleared to land because the airport is private and that I had to declare an emergency if I wanted to land; due to that I applied full power in order to divert to ZZZ2 and that caused engine roughness to increase. I decided to [advise ATC] and land on Runway XX at ZZZ. After the aircraft landed; I inspected the aircraft for any possible leaks or abnormal damage but there were none. None of the occupants suffer any injuries or the aircraft damage. In the right front seat was a student pilot and in the back seat; a Private Pilot (as passengers).