I feel like various iterations of this question have been asked and I have been through all of them without a solid answer. I'm a Windows user and have the latest versions of everything installed as of the posting date.
I previously only used Lightroom Classic with all of my photos locally stored. I successfully migrated my catalog and all of my photos appear in Lightroom as well as Lightroom Web. I was running out of local storage so this was a must. I don't care to have any photos stored locally. Here are my issues.
1) Can Lightroom Classic work at all if I don't store photos locally?
2) If I upload photos to Lightroom, they do not appear in Lightroom Classic
3) I have sync turned on in Lightroom Classic, but what is it syncing if my catalog has already been migrated? I want to be sure I'm not duplicating anything in my Creative Cloud storage.
There are a few things you need to understand about the catalog migration function. It was designed by Adobe to be used as a one-time activity for those users wanting to switch permanently from Classic to Lightroom. It was not intended or expected that the users would, after migration, re-enable syncing in the migrated Classic catalog.
The reason for that is that the consequences in Classic, after re-enabling syncing, are unpredictable. As @Rob_Cullen has already explained, ALL images stored in the cloud will also download into the synced Classic catalog for storing on the local hard drive(s). In the situation of re-enabling sync in the migrated Classic catalog, you are depending on Classic's ability to recognise that the new images that the cloud is trying to sync to Classic do already exist in Classic, and if it does then it should not download the images again and instead simply flag all the existing images as "synced". However, as I said, that is unpredictable (I'm pretty sure it depends on what was synced before the migrartion utility was run), and as a result you may find that some images are downloaded again, sometimes as full duplicates of the originals, but sometimes as virtual copies. Either way it can be a mess to sort out.
The first thing you need to establish is whether you have a problem or not. Start with looking at the numbers: compare the total number of images in the cloud (look at the photo count for All Photos in the LrWeb app for the accurate figure), with the number of images in Classic (two numbers here, All Photographs in the Catalog panel, and immediately below that the All Synced Photographs total). If all 3 totals are the same you are in good shape, if only the All Photos total in LrWeb is the same as the All Photographs total you are still in good shape (if All Synced Photographs total is lower, that probably means that there are some videos among the images in the cloud, which are not included in All Synced Photographs as videos do not sync even though they would be downloaded initially).
Another thing to understand is that you cannot avoid the situation of some original files being stored locally, even if you abandon Classic completely and just use Lightroom. The fact is that Lightroom itself will automatically download a copy of the cloud-based originals when doing certain functions in the Lightroom Desktop app (i.e. editing, or even just zooming to 1:1). For those functions a copy of the file is downloaded from the loud and stored locally, though if you have the "Store a copy of all originals...." unchecked those local copies should be deleted automatically over time. But you cannot avoid some local copies making their way onto your local system from time to time.