Some people prefer to speak of being "considerate to" people and "considerate of" their feelings, need for quiet etc.
People who like their vocabulary nicely segregated would into separate rôles would likely favour this clean separation, and argue therefore it was "the correct way", but language isn't always so neat.
Some people prefer to speak just of being "considerate of" both people and their feelings etc.
People who like their language use to be etymologically defensible would likely favour this clean separation, as when we consider the earliest senses of considerate (well-considered, careful, deliberate) then by being considerate in this sense "of others" we narrow down that original sense to arrive at the sense it is most commonly used in today. They might argue therefore it was "the correct way", but language isn't always so consistent etymologically. (If it were, words would never change, and there wouldn't be any etymology).
It's amusing that both have a vaguely reasonable claim to being "the only one that makes sense" to two different ways of arguing about what is correct.
Most people of course just use what they heard growing up. I would have been sure that "considerate to others" was the more common, but ngrams suggests that in fact "of others" is.
But in any case, both are very definitely correct, and the person who altered the sign should have considered examining usage first.