Can I share an unpublished paper without this counting as prior publication?
As per your comment, your main concern is that your sharing the draft may count as prior publication and will prevent you from eventually publishing the final paper.

Sharing a draft does not count as publication.

Moreover, your draft is most likely going to change a lot before you submit the final manuscript to a journal for review. In general, it is a good idea to share drafts early and often, so you can collect as much feedback as possible. 

Some precautions:


make the draft status explicit 
ensure the paper can't be found online before acceptance, so as not to compromise blind review (may depend on the field)
ask that the draft paper not be cited or circulated without permission