In the primordial ocean the first compounds were synthesized essentially by random chance using available energy sources, including radiation and lightening; here chirality was 50/50.
From this "soup" the first, very rough, self-replicating "things" emerged.
These thrived consuming the large amount of available organic material.
As soon as these materials become less abundant a strong evolution push appeared: the ability to produce needed complex (rare!) compounds from relatively abundant "simpler" ones.
This lead to first enzymes acting as catalysts for chemical reactions.
This is the point where symmetry was broken. Most (if not all) enzymes catalyze a single chirality.
This means (relatively) shortly after first "true" living proto-cells started colonizing the ocean (much before fitoplancton started changing atmosphere composition) the vast majority of organic compounds available were produced by "living" cells... and thus with a specific chirality.
It is conceivable two non-communicating primordial oceans could have given birth to unrelated bio(emi)spheres with partially (or completely) different compounds, possibly with different chirality. As a very special case it would be possible the two bio(emi)spheres could include the same compound with different chirality.
Probability of something like this happening in practice is anyone guess, but I regard it as rather low. Rationale being a very long and complex chain of unlikely events was needed to produce living cells in the first place and chance this chain could happen twice in the time span needed for first one to fill all "likely" environments is quite low.
Complex reactions are very difficult to set-up and replicate; as an example mitochondria (who learned how to produce energy from sugar and Oxygen) were never replicated, but accepted as symbiotic "partners" inside other cells, being so successful there's virtually no cell not hosting them (and they are all closely related; no "parallel evolution").
Bottom line: it is conceivable to have different and unrelated biospheres on the same planet, given suitable morphology. It is also conceivable these two biosphere could come in contact much later (perhaps because of a tectonic movement removing the barrier). In this case, however, it's extremely unlikely the biota would be anywhere near "similar". Chirality would be the least of the problems.
O.P. asked: At some point the progeny of these two incubators – whether two oceans a world apart or two tide pools metres apart – have to come in contact. Then what happens?
It (as always) depends on specifics. I'll attempt a speculation.
In general there will be an initial stage where the two different populations will simply ignore each other and compete on an equal basis for the available resources. Compounds produced by the "other world" would be useless or even toxic (because the enzymatic set isn't able to handle them) and thus no direct interaction will happen.
From this moment, as said, the two populations will be in competition for resources, this means if one has a significant edge over the other then it will slowly drive the "less efficient" "half world" toward extinction (it may resist in some sheltered niche). This is what happened to anaerobic bacteria in our world when more efficient aerobic metabolism appeared.
OTOH, if efficiency (even if based on different chemical processes) is about the same then some kind of "mutual recognition" is going to take place.
There will be a push to:
- avoid poisoning from the "other"
- find ways to metabolize "alien" compounds.
Each step in this direction risks breaking the above-mentioned equilibrium and will push to develop "counter-measures". Something similar to "invention" of lignin by trees which took million years to bacteria to find out a way to cope with.
Here you can imagine two divergent results:
war: competition to death and thus one of the two "halves" prevailing sooner or later and driving the other to extinction (or to some biological enclave).
cooperation: development of one or many symbiotic arrangements where complex biological interactions will include species from both "halves".
In any case it would be extremely unlikely "parallel evolution" of any kind.