Which is why a common anti-vax argument is sow doubt about the harmful effects.
I noticed a few options:
a) conspiratorial: "the government is lying about side harm"
b) anecdotal: "I personally know at least 3 people who couldn't walk after the vaccine. Do you call me a liar?"
c) hyper-skepticism: "how can we know if it long term side effect? Yes it's proven safe for X months, but what about in Y month? what about Y years?"
I found it hard to defuse such positions with "just more data". They seem to be memes (crafted or just evolved?) that resist quite strongly to data.