Hi Matt,
Unfortunately I'm not seeing a way to do this with the wiki. I highly recommend putting this in as a request in the Suggest Features forum, as other folks could certainly use this option as well.
Depending on how complex the structure needs to be for this, you could potentially use the Assignment with the Online Text submission option to allow learners to add content that is private from their peers but visible to the Trainer. Assignments do allow the Trainer to do a bulk download and you can leave the assignment open so that the learner can keep coming back and adding to their work.
Thanks!
Wes