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!