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I am looking for an indigenous Russian word for women who entertained guests at social gatherings in Russia. To put it simply, I am looking for a Russian analogue of geisha.

I tried to find such a word on my own, but found only an intriguing statement in Wikipedia:

Вместе с тем для обозначения подобных артисток и их профессии, как в японском, так и в остальных мировых языках, включая русский, встречаются и другие наименования. (Link)

This statement says there is a Russian word for that, but does not name that word.

Тамада.
It can be a woman or man. This person entertains guests at social gatherings like wedding, parties, in Russia.
It reflects only this part of geisha.

I'm not aware of a native Slavic word for this profession.

Since this profession has come to Russia from the West, Russian uses loanwords to describe it: куртизанка, кокотка, демимонденка, гетера, эскортница etc., including гейша itself.

Note, however, that all those words (except гейша naturally) don't describe a woman who would do exactly what a geisha would do.

That's why we all know the word "geisha" in the first place: if we had an exact native word for a geisha, we would not need the Japanese word.

Вместе с тем для обозначения подобных артисток и их профессии, как в японском, так и в остальных мировых языках, включая русский, встречаются и другие наименования.

I suppose this piece of text means "there are some other words for geisha", i. e. "other words for this Japanese profession" because there is no such profession in Russian culture.

Nowadays, there is a word with a similar meaning - "аниматор". Usually "аниматор" is a person who entertains children on birthdays. Also, hotels in Turkey which work on "all inclusive" model, have animators for adults.

UPD

As for the word "массовик-затейник" proposed by Quassnoi in comments, it is a word from the Soviet epoch, which means more or less the same as "аниматор".

"Тамада" is a man or a woman, who entertains guests at weddings, bithdays and so on, organizing contests and games. Usually it takes place at weddings and anniversaries in restaurants.

"Эскорт" is a woman (or a number of them) who are going to different places with somebody who pays them, but actually their job usually (not always) includes intimate services as well. There are differencies between "эскорт" and prostitutes, you can read articles about it.

"Аниматор" is a someone who entertains you (with contests, games, quests, etc.) and you pay them only for that. Usually "аниматоры" are being paid to entertain children when their parents are eating in restaurants, relaxing at the seaside and so on.