Out the exhaust pipe?
Hopefully as an almost-invisible blue haze. Get it wrong and it's dripping out the exhaust, and covering your boots and trousers and the back half of the bike as it spews out from every conceivable joint.
The presence of the built-in return pump makes your engine "dry sump" rather than "total loss" (or "wet sump" as Haycraft rather misleadingly calls it in his Book of the JAP Engine). In principle the oil goes back to the oil tank after it's had a lap through the engine.
There were many JAP oiling systems, but 1932 was the first year of dry sump: first with a double-acting Pilgrim, then after with a built-in double acting JAP pump built in. I guess the racy engines were a bit weird. What year's you engine?
You'll be able to see the oil going into to the engine through the window on top of the Pilgrim pump. Make sure you understand where it goes next: as it goes into the end of the timing-side main shaft there is often a seal of some kind (e.g. a sprung brass quill) which can be missing, and then an oilway should be drilled all the way to the big end. Use your oil can to give a good squirt of oil into the main shaft - it should come out of the big end bearing. If not, double check with your JAP engine man. (I bet he checked!) The big end will die first without oil, so it pays to be certain.
Have fun!
Leon
Edit: OK, I've been back through the thread (not much of it!!!!) and found photos on p7 that show the oil layout. There's no return pump per-se? Just the oil pushed through the flappers into the small chamber underneath, with the neck at the back where the oil exits. Onto the dirt track originally, but better just dump it into a catch bottle for the road. What a waste of good oil!
Not sure of the exact year of the engine, since (afaik) no two pieces came from the same "factory" engine: it's a complelye "bitsa" scrounged and found. But it's essentially a late 1940s / early 1950s flat-track/speedway engine in intent, but detuned (350 crank , 500 conrod, piston, barrel, etc).
It's worth remembering that the old speedway/flat-track race engines cared a lot more about getting power from the unit than worrying about the oil loss, and on shale / grasstrack it's not going to matter if you're losing a half a litre of oil per race, you fill up for the next one! My thinking is that ny modification for catching the lost oil have come from the Formula 500 car racers who were using these engines on proper race-circuits and so needed to a) reduce droppage for safety reasons, and b) reduce costs.
These photos show the oiling system I have on this bike:


Pilgrim Pump mounted to a secondary pump, mounted to the Timing chain cover. The oil comes from the tank (the outboard-most pipe into the back of the Pilgrim), fed into the engine and the head, then is "sucked" (well - I doubt there's a pressure difference...) by that pump which is inboard of the Pilgrim from the "sump" (ie, the exit of the sludge-trap) and returned up to the oil tank.
My father, when he built this bike, was very aware of the total-loss nature of the engine as it would have been originally, but he wanted a bike he could use on the road, and on classic trackdays. So, he spent a
lot of time getting this "concept" (ie returning the lost oil back to the tank) to work. His original idea was to have the oil drain from the "sump" down to a lower mounted tank, and then pumped up to the oil tank via a modified bicycle pump which was activated by the movement of the swingarm.
It actually worked. Well - on bumpy roads anyway.... it turned out that modern race-tracks are too smooth and there wasn't enough movement of the swinging arm to make the pump work.

He also went through a couple of variations on the theme of multiple catch-tanks, but none were very satisfactory.
So he tried a few other ideas and eventually found this secondary pump and the engine had this fitted for... the last few years (at a guess) of it's active life.
I do need to check the oil-flow down the crank, and look into whether or not I need that sprung quill: I've seen it on the list of JAP items from Speedway Service, but pretty sure there wasn't one fitted to this engine when I stripped it. Something to ponder and discuss with the expert.