When I studied in St. Petersburg for a bit, our teacher told us that живот derives from жить/жизнь (to live/life), because a long time ago it seemed that pregnant women carried their babies in their stomach.
I got the impression so far that when Russians speak about their own body they tend do drop the possessive pronouns.
Given that, I think if someone else's stomach were implied, a possessive pronoun would have been used to indicate the fact
In spanish, "sleep on his stomach" is "dormir bocabajo", that is, literally, "sleep mouth-down". The opposite would be "dormir bocarriba" or "sleep mouth-up". Today I've learnt how it is said in russian and english.
(I prefer Russian for Spanish speakers, but not as much as i prefer the app to the web page. (No beta.)
— My first thought was how the idiom was the same in US English & Russian. (Once again, sorry Brits.)
"to sleep on stomach" is a set expression meaning the person sleeps facing the ground
Likewise in English there is "sleep on the side" and "sleep on the back", which coincide with the Russian "спать на боку" and "спать на спине" respectively.