The threshold isn't defined. Reputation and influence alone is a pretty broad requirement. For example, Luffy would certainly qualify as a "great pirate" right now. He has influence and deeds under his belt. But something's missing.
Imo, you have to have accomplished something which has forever changed the world. We know that Roger defeated Rocks alongside Garp, but the rest of the world did no, so we'll exclude that. Looking at Newgate and Linlin, they had control over countless territories, controlling great parts of the New World all on their own. There was virtually nowhere someone could go without having their influence felt. They have a firm foothold within the World of One Piece.
Luffy's almost there, but not quite. He's been involved in a lot of world shaking events, and the media has taken notice of it, but there's still that uncertainty behind him. A legitimate rising star or an upstart who will have his wings' clipped by the more established Emperors? Those kinds of discussions aren't present with regards to the other Emperors.