(1) Background: Health workers (HWs) are at high risk of acquiring SARS-CoV-2 (Severe Acute Respiratory Syndrome Coronavirus 2) infections. Therefore, health authorities further recommend screening strategies for SARS-CoV-2 infection in exposed or high-risk HWs. Nevertheless, to date, the best/optimal method to screen HWs for SARS-CoV-2 infection is still under debate, and data on the prevalence of SARS-CoV-2 infection in HWs are still scarce. The present study aims to assess the SARS-CoV-2 infection rate amongst HWs in a teaching hospital in Central Italy and the diagnostic performance of SARS-CoV-2 serology (index test) in comparison with the SARS-CoV-2 RNA PCR assay (reference standard). (2) Methods: A cross-sectional study on the retrospective data of HWs tested for SARS-CoV-2 by RNA-RT-PCR on nasopharyngeal swabs and by an IgM/IgG serology assay on venous blood samples, irrespective of exposure and/or symptoms, was carried out. (3) Results: A total of 2057 HWs (median age 46, 19–69 years, females 60.2%) were assessed by the RNA RT-PCR assay and 58 (2.7%) tested positive for SARS-CoV-2 infection. Compared with negative HWs, SARS-CoV-2-positives were younger (mean age 41.7 versus 45.2, p < 0.01; 50% versus 31% under or equal to 40 years old, p < 0.002) and had a shorter duration of employment (64 versus 125 months, p = 0.02). Exposure to SARS-CoV-2 was more frequent in positive HWs than in negatives (55.2% versus 27.5%, p < 0.0001). In 44.8% of positive HWs, no exposure was traced. None of the positive HWs had a fatal outcome, none of them had acute respiratory distress syndrome, and only one required hospitalization for mild/moderate pneumonia. In 1084 (51.2%) HWs, nasopharyngeal swabs and an IgM/IgG serology assay were performed. With regard to IgM serology, sensitivity was 0% at a specificity of 98.99% (positive predictive value, PPV 0%, negative predictive value, NPV 99.2%). Concerning IgG serology and irrespective of the time interval between nasopharyngeal swab and serology, sensitivity was 50% at a specificity of 99.1% (PPV 28.6%, NPV 99.6%). IgG serology showed a higher diagnostic performance when performed at least two weeks after testing SARS-CoV-2-positive at the RNA RT-PCR assay by a nasopharyngeal swab. (4) Conclusions: Our experience in Central Italy demonstrated a low prevalence of SARS-CoV-2 infection amongst HWs, but higher than in the general population. Nearly half of the positive HWs reported no previous exposure to SARS-CoV-2-infected subjects and were diagnosed thanks to the proactive screening strategy implemented. IgG serology seems useful when performed at least two weeks after an RNA RT-PCR assay. IgM serology does not seem to be a useful test for the diagnosis of active SARS-CoV-2 infection. High awareness of SARS-CoV-2 infection is mandatory for all people, but especially for HWs, irrespective of symptoms, to safeguard their health and that of patients.