If you get a rejected dispute you can email copy...@youtube.com and request that they forward you the contact details of the claimant. They will generally only release the contact email for the claimant after the dispute has been rejected.
If you are just waiting for the dispute to be released, and it has not been rejected by the claimant, they probably won't give out the contact details. In these cases the claim will automatically expire if the claimant does not reject the dispute within 30 days. The expiry date appears on the dispute status page.