The second is not slower for the someone walking, and it's not slower for the person standing still. What is slower is the way the standing person conceptualizes a second for the walking person-- the way the standing person matches up mentally the beginning and end of his/her second to the beginning and end of the walker's second. The way to tell this must be true is to note that the walking person thinks it is the standing person's second that matches up to be too long. In the presence of acceleration or gravity, like in the twin paradox, you can make the two come together, and only then will they have to agree on the same "matching up", which will also end up exhibiting less elapsed time for one than the other. But that doesn't mean anyone's "time was slower", it just means you ended up matching up less time to more time between the two.