WRT the UK, Connelly v DPP  1 ALL ER 1012, it was found that
The words “grossly offensive” and “indecent” are ordinary English
words. They are not used in a special sense in section 1 of the 1988
In this instance, the lower court decided that her actions were grossly offensive; for her to win her appeal (which she did not) she would have to show that the decision was "one which no court acquainted with the ordinary use of language could have reached". The higher court agreed that the photographs were "shocking and disturbing". The Communications Act 2003 section 127 likewise does not define "grossly offensive", meaning that it has it's ordinary meaning. Something is offensive if people are offended by it (the number or nature of the people so offended is not part of the meaning of offensive), and "grossly" is an intensifier suggesting that many people are offended, or people are highly offended. In other words, it refers to an emotional reaction. Whether or not a person has that reaction depends on the subject of the emotion (the person having it).
Freedom of speech is protected to various degrees, by country. For example, blasphemy against Islam is punishable by death in Saudi Arabia, but not the US. It is a crime to criticize or insult the royal family in Thailand, but not the UK. The term "obscene" is often used to focus laws on sexually-offensive material, and in the US you don't generally find laws against "offensive" speech (I don't know of any survivors, though there may be some).