It seems sometimes "our jackets" is acceptable for this kind of sentence and sometimes not. Is there a rule of thumb, or is this a duo thing?
No, Duo is wrong here. Both "the jackets" and "our jackets" should be accepted here. In these kinds of statements Spanish often uses the article rather than the possessive that we'd use in English.
The suggested correction is "In summer we do not put on thejackets." There should be a space between the and jackets. No option to report this.