I have a workflow on a document library. If I update that workflow in sharepoint designer and save, is it necessary to do a publish for that change to "take effect" in sharepoint, or is it enough to save it, since the workflow is already published?
You have to publish the workflow in order for it to take effect.
Saving is just for persisting the changes you've made so you can get back to them (you can also save it while it has validation errors).
You have to publish it in order to disable the old version and enable this new one.