(1) 'Hospital' and 'school' without the article applies to patients and students only (I believe-- I need BrE corroboration on this). (2) In your sentences, 'the' seems a bit unusual, and would more normally be replaced by 'a' when identifying their places of employment. (3) As to your main question: either preposition is OK, but 'at' is more expected, as we are considering a point location-- a type of facility-- rather than the building itself, which they would be inside or 'in'.
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