With a game like The Elder Scrolls: Skyrim, the legendary edition is just a copy of the original game, with all of the DLC. A good way to check this out in the future is to look at the achievements page (trophies for PlayStation).
If the legendary edition has a different achievement page (may even be the same achievements, just a different entry), save transfer is questioned as they are not the exact same game.
For games where you are just getting DLC content with the new version, you are almost always playing the same game in terms of your save content. Games like Diablo 3:Reaper of Souls are entirely separate games (often due to extensive game change) and do not naturally exchange save data - however, developers may put work arounds in, such as the "character export" feature of Diablo 3.