How does information consumption affect behaviour in the context of the COVID-19 pandemic? A popular hypothesis states that the so-called infodemics has 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. 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 find that such a tumultuous information landscape translated in only minimal variations in vaccine acceptance as measured by Facebook's daily COVID-19 World Symptoms Survey on a sample of 1.6M users. This finding supports the hypothesis that altered information consumption patterns are not a reliable predictor of behavioural change. Instead, wider attention on social media seems to resolve in polarisation, with the vaccine-prone and the vaccine-hesitant maintaining their positions.