Immunologist in training here. The answer is we simply do not know yet. We don’t have enough evidence to know exactly how long immunity lasts after natural infection. Some experts say a couple of months. Usually the first of any vaccine induces a slightly dampened immune response (in cases where a booster is needed). This is because your body is making antibodies towards a foreign invader for the first time. During the second vaccine (booster), people typically have a more pronounced immune response because your trained immune cells come out to fight with antibodies in hand. In other words they will remember your first encounter with this invader. During this time, your body makes molecules (known as cytokines and chemokines) in much higher levels to help the immune response which are responsible for the side effects. However everyone’s immune system is so individualistic, we react in slightly different ways to all invaders. In fact, some do not react at all. Although I have not seen any data, I would suspect the response would be dependent on how long after COVID infection the booster was given. They are recommending that if you have COVID, you should wait 14 days to be vaccinated. This is for two reasons, A to prevent the spread at vaccination sites and B to generate a stronger response to the vaccine. I would agree with you that in general, someone that had their first dose and became naturally infected would probably have more of an immune response but the infection might be cleared faster and symptoms would not last as long. However, there are people who are asymptomatic to both vaccination and natural infection.