"I am Italian and I am working in the US as a researcher. They are telling me that The institute I work for is not directly linked to any university and cannot be considered an educational institution. It's also not a medical facility. Therefore I am not qualified for tax exemption.Still, I am not working for a company and I am here as a postdoc researcher with a J1 Visa. As stated by the treaty :'The exemption does not apply to income from research carried on mainly for the private benefit of any person rather than in the public interest'. I don't believe this applies to my situation. Am I exempted for the first two years or not? Thank you

The treaty is clear in stating that the sponsoring organization MUST be an educational institution.

If your sponsor specifically states that they do NOT qualify as an educational institution, I would be reluctant to argue with them.