I've researched a little online and stumbled upon a few interpretations of the origins of oss but most of them seem to tackle it the wrong way. There are actually two separate words that seem to be mingled into one but that most Japanese seem to differentiate.
The oss mostly used in martial arts is actually said differently from the other word 'ss. 'ss starts with a muted exhaling sound instead of an open-mouthed "o" use in the oss followed by the same dragging "ss".
osu → oss
Oss, as explained in other articles comes from 押忍, as used in sports in general (baseball students or any other team that trains seriously tend to use it here in Shikoku island) starts with an open mouth sound and finishes with a dragging "ss".
...masu → mss → 'ss
The other similar word 'ss comes from an abbreviation of other words. Those words can be shortened differently depending on the situation and urgency or roughness. Roughness usually equates to determination for men/boys who shorten osu or the other words and is always said with conviction. I've never heard a half-assed 'ss or oss... yet ;)
- Onegaishimasu → Onuhashmass → uhmass → mss → 'ss
- Arigato gozaimasu → agmassu → agmss → mss → 'ss
- Ganbarimasu → ganbamassu → gamss → mss → 'ss
- Yoroshiku onegaishimasu → yoroshknegaishmass → yoshmass → mass → 'ss
P.S.: I was surprised that no articles mentioned the word ganbarimasu that is widely used all around Japan and shortened the same way.
Compared to the osu, two of those words are often used in the past tense and that's where they become easy to differentiate since there is no past tense for oss.
- arigato gozaimashita, ganbarimashita → ... [various shortened versions] ... → 'shta*
*Quite a few other words are shortened to 'shta since it's the common past tense conjugation.
Students and teachers using oss everyday sometimes replace the common 'ss/'shta with an oss in many situations but not in all of them. People not using oss would normally 'ss or 'shta unless they want to show ultimate determination by borrowing the oss from competitive sports. It is then a little out of place, since people around would know that the person isn't involved in an activity usually using oss, and would have a stronger impact.
I guess it's a risky thing to use oss for someone not able to "read the mood" and the oss might be perceived as a joke instead of determination if not said with enough seriousness.
I think that Japanese can put up with a lot of misuses of the language as long as it's not overused or continuous. I don't think that the general public would take any offense in using oss but since it's usually serious business, the best thing is to look at the Japanese people present while saying it and see if they look irritated. Once would be fine anywhere... but Japanese expect people to realize it when they are annoying; reading body language/facial expression is a necessity in Japan.
I hope my input helps to clarify the origins of oss and its close neighbour 'ss. I am sure that if you start to listen carefully you'll "see" the difference quite fast.