Sparrow uses someone else's node by default, but yes if you are running core, you can connect Sparrow to core to use your own node (I don't know anything about Wasabi, but I would guess its the same thing)
Sorry I might of confused wasabi and sparrow! Wasabi can run bitcoin knots local node (is a setting), sparrow just requires you to run bitcoin core at the same time and to also check a setting. So not completely out of the box, bit still a damn sight better than wallet manufacturer’s default software.