If a draft is created by the Create draft automation action by Contractbook in-app automation builder, the automation creator will be automatically assigned as the owner of the new draft, disregarding if the chosen in the automation trigger event template is one of Contractbook premade templates or a template shared with you instead of one you own yourself.
In case you need to change this generic setup, you can do that by creating an additional automation step Share document right after the Create draft automation step.
In this extra automation step, you can grant any collaborator full edit rights and privileges to send the new draft for signature. They will then have almost the same permissions as the document owner themselves. The only limitation will be that they will not be able to see the Version History of the draft.