How information consumption affects behaviour is an open and widely debated research question. A popular hypothesis states that the so-called infodemic has a substantial impact on orienting individual decisions. A competing hypothesis stresses that exposure to vast amounts of even contradictory information has little effect on personal choices. The COVID-19 pandemic offered an opportunity to investigate this relationship, analysing the interplay between COVID-19 related information circulation and the propensity of users to get vaccinated. We analyse the vaccine infodemics on Twitter and Facebook by looking at 146M contents produced by 20M accounts between 1 January 2020 and 30 April 2021. We find that vaccine-related news triggered huge interest through social media, affecting attention patterns and the modality in which information was spreading. However, we observe that such a tumultuous information landscape translated only in minimal variations in overall vaccine acceptance as measured by Facebook's daily COVID-19 Trends and Impact Survey (previously known as COVID-19 World Symptoms Survey) on a sample of 1.6M users. Notably, the observation period includes the European Medicines Agency (EMA) investigations over blood clots cases potentially related to vaccinations, a series of events that could have eroded trust in vaccination campaigns. We conclude the paper by investigating the numerical correlation between various infodemics indices and vaccine acceptance, observing strong compatibility with a null model. This finding supports the hypothesis that altered information consumption patterns are not a reliable predictor of collective behavioural change. Instead, wider attention on social media seems to resolve in polarisation, with the vaccine-prone and the vaccine-hesitant maintaining their positions.