The audio is very accurate. You can also check the pronunciation on forvo.com - here it's text-to-audio but on forvo it's native speakers. As you'll probably notice in other discussions, this sound, which non-native speakers often pick up as sh, is not a sh. In fact, Greek native speakers deny passionately that such a sound even exists in Greek. It is actually something between a s and a sh, and -apparently- also exists in Spanish, if that helps. Unless you can reproduce that half-way sh, you should only aim for a simple s sound, because sh sounds plain wrong to a native's ears. :)
Edit: this sounds comes up a lot, e.g. the final s in words ending in -ος or ως, e.g. ωκεανός.
Thank you for your quick and helpful answer. It happens I am a French man living in Spain (Valencia), so that my French ears are used to make a difference between the two phonemes /s/ and /sh/ but they are also used to recognize the one Spanish phonem /s/ phonetically realized somewhere between the sounds [s] and [sh], at least in Spain [it's not true in American Spanish] (sorry, I don't have the true IPA symbols on my mobile).
I checked on Forvo, and what I heard was very close to the Spanish /s/ (not exactly the same though). But when I listen to the audio here, what I can hear is much closer to the [sh] sound than what I can hear listening to native speakers on Forvo. The fact is this TTS audio is not shocking for your Greek ears, which means it's not a true [sh] sound and my French ears are wrong, but in a blind test, it's likely I would recognize a native speaker saying ψομί from the Duolingo TTS engine saying the same word.
I think it won't be a problem for me to imitate the Greek /s/ sound as I can hear it from native speakers. The problem will be more to immediately identify the Greek /s/ sound when listening it, without having to think "what I hear is a [sh] sound, but it doesn't exist in Greek, so it must be a Greek [s]"!
I can't comment on the Japanese which I am not familiar with. I'd say your conjecture that "they are independent sounds because of the accent" is accurate.
Here are two sites with pronunciation.
https://forvo.com/search/%CF%84%CF%81%CF%8E%CF%89/ This has native speakers. (Careful there is also Ancient Greek which you don't want here.)
http://www.oddcast.com/home/demos/tts/tts_example.php This is a TTS rather like the one we use.
I should point out that the normal speed pronunciation shown with this sentence is correct but at slow speed, it is really odd.
I think that εγώ τρώω ψομι means I am eating bread and τρωω ψωμι means I eat bread. Am I right?
Bread is always spelled ψωμί (with an omega, not an omicron).
And εγώ τρώω is the same as τρώω -- either can be translated as "I eat" or "I am eating": this is not a difference that Greek has in its grammar.