Anecdotal evidence is filtered evidence.

Right, the existence of the anecdote is the evidence, not the occurrence of the events that it alleges.

You can find people saying anecdotes on any side of a debate, and I see no reason the people who are right would cite anecdotes more.

It is true that, if a hypothesis has reached the point of being seriously debated, then there are probably anecdotes being offered in support of it. (... assuming that we're taking about the kinds of hypotheses that would ever have an anecdote offered in support of it.) Therefore, the learning of the existence of anecdotes probably won't move much probability around among the hypotheses being seriously debated.

However, hypothesis space is vast. Many hypotheses have never even been brought up for debate. The overwhelming majority should never come to our attention at all.

In particular, hypothesis space contains hypotheses for which no anecdote has ever been offered. If you learned that a particular hypothesis H were true, you would increase your probability that H was among those hypotheses that are supported by anecdotes. (Right? The alternative is that which hypotheses get anecdotes is determined by mechanisms that have absolutely no correlation, or even negative correlation, with the truth.) Therefore, the existence of an anecdote is evidence for the hypothesis that the anecdote alleges is true.