From Wikipedia, it says the etymology is from the "Montagnais word meaning "snowshoe-netter", and the Inuit Circumpolar Conference used both Inuit and Eskimo. So apparently you are allowed to say Eskimo without offense.
Apparently, "Eskimo" is a Red IndianNative AmericanIndigenous American Algonquinian word meaning "raw-flesh eaters", a term one might justifiably use for people who enjoy sushi or steak tartare, so I am not sure why it would be offensive. Certainly, when I was growing up in Baltimore in the 1960s and 1970s, I never heard used in any way that was intended to cause offense. When I lived in Canada in the 1980s, I learned that the indigenous dwellers of the Canadian Arctic preferred to be known as Inuit, while those who lived in Alaska preferred to be called Eskimos, at least by non-Inuktitut-speakers. Perhaps this was because they didn't want people to botch the pronunciation of their ethnonym or because it got on their nerves when someone referred to one of them as "an Inuit" since "Inuit" is a plural form and this should properly be "an Inuk". In any case, there was no suggestion that "Eskimo" was offensive, just that it was not what they called themselves. It annoys me that people get so sensitive when it is clear that no offense is intended. Is it insulting to refer to Angela Merkel as a "German", for example, when she calls herself a "Deutsche"? Should we refer to Japanese people as "Nihonjin"?
I'm not convinced that eskimo is insulting to Inuit, although it's not particularly my area of expertise. Here there is a perception among the wider community that Aborigine is derogatory. I have heard quite a few tell me that they are proud to consider themselves Aborigines rather than indigenous Australians or any of the other terms more in vogue.