The vaccines against COVID-19 arrived in Spain at the end of 2020 along with vaccination campaigns which were not free of controversy. The debate was fueled by the adverse effects following the administration of the AstraZeneca-Oxford (AZ) vaccine in some European countries, eventually leading to its temporary suspension as a precautionary measure. In the present study, we analyze the healthcare professionals’ conversations, sentiment, polarity, and intensity on social media during two periods in 2021: the one closest to the suspension of the AZ vaccine and the same time frame 30 days later. We also analyzed whether there were differences between Spain and the rest of the world. Results: The negative sentiment ratio was higher (U = 87; p = 0.048) in Spain in March (Med = 0.396), as well as the daily intensity (U = 86; p = 0.044; Med = 0.440). The opposite happened with polarity (U = 86; p = 0.044), which was higher in the rest of the world (Med = −0.264). Conclusions: There was a general increase in messages and interactions between March and April. In Spain, there was a higher incidence of negative messages and intensity compared to the rest of the world during the March period that disappeared in April. Finally, it was found that the dissemination of messages linked to negative emotions towards vaccines against COVID-19 from healthcare professionals contributed to a negative approach to primary prevention campaigns in the middle of the pandemic