For more than a decade, doubt about vaccines has become an increasingly important global issue. Polarization of opinions on this matter, especially through social media, has been repeatedly observed, but details about the balance of forces are left unclear. In this paper, we analyse the flow of information on vaccines on the French-speaking realm of Twitter between 2016 and 2017. Two major asymmetries appear. Rather than opposing themselves on each vaccine-related controversy, pro and anti-vaccine accounts focus on different vaccines and vaccine-related topics. Pro-vaccine accounts focus on hopes for new groundbreaking vaccines and on ongoing outbreaks of vaccine-preventable illnesses. Vaccine critics concentrate their posts on a limited number of controversial vaccines and adjuvants. Furthermore, vaccine-critical accounts display greater craft and energy, using a wider variety of sources, and a more coordinated set of hashtags. This double asymmetry can have serious consequences. Despite the presence of a large number of pro-vaccine accounts, some arguments raised by efficiently organized and very active vaccine-critical activists are left unanswered.