Like many international markets, the crude oil market is a mix of private and public participants.
From most perspectives, it really doesn't matter whether oil production is state-run or run by private leases. Both, ultimately, are combinations of public and private forces - just different blends. Both are trying to optimise the outcomes for their owners. They're all suppliers that respond to demand.
They're all largely price takers; the larger they are, the more they contribute to being a price maker - same as any other market. Nominally, Saudi Arabia as the main force in OPEC carries the market force of the entire cartel: but as Saudi is by far the largest producer in OPEC, the supply levels of the other cartel members don't carry anywhere near as much weight, and there is some evidence that they haven't always implemented fully proportional cuts when Saudi has called for cartel-wide production cuts.
So Saudi Aramco is just another market player, albeit the single largest one, and has some additional leverage via other OPEC members.
Many myths surround Saudi's oil policy; many widely-held assumptions are wrong. The most accurate thing I've heard said of it, credited to various folks is: the people who talk about Saudi intentions, know nothing; and the people who do know the intentions, say nothing.
Footnoe: the products are not completely fungible: different crudes have different chemical compositions (sweet crude has low sulphur content, sour crude has high)