Doesn't the fact that a large percentage of people are already vaccinated with another covid-19 vaccine effect these results? I know if there is a control group

Doesn't the fact that a large percentage of people are already vaccinated with another covid-19 vaccine effect these results? I know if there is a control group you can compare the rate of infection in the two groups but it is my understanding that when they say a vaccine is 96% effective they simply mean if 1000 people are given the shot 4 percent of the people got covid during the study, right? So if many of the people who are around the trial subjects are vaccinated would this not lead to inflation of the "effectiveness" of vaccines tested later when more people are vaccinated? Or do they take into account the control group results / vaccination rates when calculating the efficiency? EDIT: Ignoring of course the fact that the specific vaccine formulas used are likely to lose effectiveness over time as new variants pop up.