I used to not sift it, but at repotting I saw how much mud the fines left in the bottom of the pot. So I now sift it. I have some simple 18 inch square wooden frames that I made the first is 1/2 inch, the second is 1/4 inch and the bottom is 1/8 inch. Everything that goes completely through just goes on the ground (helps the lawn, lol), the 1/8 to 1/4 is used for bonsai soil, the 1/2 inch and up is used as a top dressing for freshly repotted trees. I used to use sphagnum moss for a top dressing, but in my neck of the woods it turn into a hot mess in a few weeks (algae and black goo). If you are repotting every couple of years, the bark will not be broken down. I have repotted some that had gone three to four years and the bark was still intact.
John