Mediumship makes claims of two kinds, and they have very different standing.
The first is a claim about experience: that practitioners undergo something distinctive, and that sitters find the encounter meaningful. That much is not seriously disputed by anyone. The historical and sociological work on Spiritualism describes a movement that gave many people, women in particular, a public platform they had no other route to in the nineteenth century, and the consolation offered to the bereaved is a real effect whatever its source.
The second is a claim about information, that a medium can state facts about a deceased person which they had no ordinary means of knowing. That is testable, and it has been tested continuously since the founding of the Society for Psychical Research in 1882, an organisation established for precisely this purpose by people who largely hoped to confirm it.
The record is long and the outcome is not favourable. Where controls have been tight, the medium blind to the sitter, the sitter blind to which of several readings is theirs, the statements scored by people who do not know their source, results have generally not exceeded chance, and the studies that reported positive effects have proved difficult to replicate. Fraud has also been found often enough to bear on the question: several of the most celebrated physical mediums of the nineteenth and early twentieth centuries were caught outright. Catching one does not settle the case against another, but it does affect how the field's own testimony should be weighed, and the investigators of the period said so themselves.
Cold reading is the standard alternative account of a successful sitting. It names a set of techniques, high-probability guesses, statements broad enough to fit most people, close attention to the sitter's reactions, and reliance on the sitter to supply the specifics and afterwards to misremember who supplied them, which together can produce a convincing reading from no information at all. It has been demonstrated deliberately, by sceptics and by stage performers who make no paranormal claim whatever. It does not demonstrate that any particular medium is using it.
The replies from practitioners deserve stating fairly. Many hold that the faculty is unreliable and will not perform to order, that laboratory conditions suppress it, and that a reading is a relationship rather than a transfer of data. Those are coherent positions and some of them are probably true. They also, followed through, place the claim permanently beyond the reach of testing, and that consequence should be named rather than left to be inferred.