Determining the presence of antibodies in serum is important for epidemiological studies, to be able to confirm whether a person has been infected, predicting risks of them getting sick and spreading the disease. During the ongoing pandemic of COVID-19, a positive serological test result can suggest if it is safe to return to work and re-engage in social activities. Despite a multitude of emerging tests, the quality of respective data often remains ambiguous, yielding a significant fraction of false positive results. The human organism produces polyclonal antibodies specific to multiple viral proteins, so testing simultaneously for multiple antibodies appeared a practical approach for increasing test specificity. We analyzed immune response and testing potential for a spectrum of antigens derived from the spike and nucleocapsid proteins of SARS-CoV-2, developed a dual-antigen testing system in the ELISA format and designed a robust algorithm for data processing. Combining nucleocapsid protein and receptor-binding domain for analysis allowed us to completely eliminate false positive results in the tested cohort (achieving specificity within a 95% confidence interval of 97.2-100%). We also tested samples collected from different households, and demonstrated differences in the immune response of COVID-19 patients and their family members; identifying, in particular, asymptomatic cases showing strong presence of studied antibodies, and cases showing none despite confirmed close contacts with the infected individuals.