This proves to be a bit difficult in your situation, as the permissions you require do not allow the user to see the ‘Owner’ field.
If each user had their own specific internal form to fill out to add contacts, the contact could easily run through an Assign Owner process in a campaign. Each internal form connected to its own sequence to assign the new contact to that specific owner, but I am not sure if this is something you would want to implement, if you haven’t yet.
I know PlusThis has an option to send contact data over, and assign an owner, based on the user that created the contact, though. https://plusthis.com/support/infusionsoft-setowner/