Interestingly enough, it's configurable both from the user side and from the referring site side.
Most (all? all the relevant ones, I think) browsers honor the referer-policy[1] header if a referring site sets it. There are options in common site frameworks, like Django[2] to control that for UAs that respect it.
Since most UAs respect it, if the indexing site had wanted to, they could easily have prevented the header from being sent for most users.