I tried to sync some photos by moving them to collections in my primary computer. They are some kind of synchronized in another computer because I can see the same number of photos in collection folder there but all the photos greyed out so I cannot do any editing there.
That option is in the import dialog for importing images, not importing a catalog. Look at the 'Changed Existing Photos' options. That is how Lightroom handles duplicates when importing catalogs. If an image has not been changed, then Lightroom won't do anything with it.