I do believe that in proper written Greek they are required (although you can surely find text without any accent marks in highly informal internet discussions, etc.).
Funnily enough, when a word is written in capital letters only, the accent mark is never used. ΥΜΝΟΣ, ΣΕΛΗΝΗ - ύμνος, σελήνη
Actually, the Greek "sigma" sounds like the European Spanish "s". However, this is not absolute, and it tends to vary by region and generation, e.g. in old Greek movies where the actors had undergone phonetics training, they pronounced the sigma like the English 's', which was considered to be the proper pronunciation. Here, I find this pronunciation to be confusing, as the final "sigma" does sound like "sh", which, to my ears as a native speaker, sounds odd.
Σ, σ is pronounced [s], a clear ordinary s, except before a. voiced consonant except λ, (i.e., β, γ, δ, μ, ν, ρ), then it is pronounced as the letter ζ (ζήτα), that is, [z]. Example o κοσμος του/ your world [o kozmos tou]
The final ς is pronounced [s] except if the next word starts with a voiced consonant, see above, then it is pronounced [z] Example o κοσμος μου/ my world [o kozmoz mou]
(Never sh, though some Greeks have unclear s-sound) See The Details of Modern Greek Phonetics and Phonology https://www.foundalis.com/lan/grphdetl.htm