Ideally the case definitions should be both sensitive and specific, but specificity is a more crucial factor than sensitivity.

However if the case definition is non-specific, then background illness may be misclassified as cases. If these false positives distribute equally between vaccinated and unvaccinated, then the AR will rise disproportionately in vaccinated compared to the unvaccinated. Remembering the equation results in an increase in RR - thus resulting in a falsely low estimate of VE.

Sensitivity is important to gain a more precise estimate. Low sensitivity lowers the power of the study, but does not alter the point estimate,  providing the sensitivity is the same for vaccinees and non-vaccinees (which we deal with under case-ascertainment) - as you lose both vacc and unvacc cases.