If someone is the reputed owner of a shop, that is what people say — but nobody has stood up and proved it. Reputed puts a soft "so they say" over a claim. Police and newspapers like it ("a reputed gang leader") because it lets them report talk without treating the talk as fact. Though actually, it is not the same as famous. A reputed expert may be well known, but the word is doing a different job: it is hedging. You can still hear an older sense, "held in high regard" ("a reputed scholar"), but the "according to rumour, not definite" sense is the one learners meet most.