"Cooperative Marks"

A "cooperative mark" is a business term, not a legal term, because the legal term is a mark, or a trademark or service mark to be more specific. My understanding of a cooperative mark is a mark that is jointly used by two or more business entities, usually for a shared or common business promotion.
Who owns it is a good question because I think the short answer is "it depends." It depends upon how the parties agreeing to use the mark arrange the ownership. It could be a joint venture, a partnership of some sort, a license or cross-license from one party to the other, or the like. What's important is that the parties cooperating to use the mark develop some sort of ownership arrangement that is properly reduced to writing by a lawyer, especially with respect to how such cooperation can be terminated at some point in time (whether or not a termination is foreseen or planned).
As far as applying to register the cooperative mark, the registrant has to be the legal owner of the mark. So if one party is to own it and license its use to the other, the licensor would apply to register the mark. If the mark is to be jointly owned, which is not a great idea for various reasons beyond the scope of this answer, both parties have to be the applicant applying to register the mark. If for some reason the trademark application conflicts with an earlier similar registered (or applied-for) mark that is owned by one of the cooperating parties to the mark and which the examiner cites in the examination of the application, the owner of the earlier mark could always issue the subsequent trademark applicant a "consent to register" which that applicant can submit to the USPTO.
If you need additional information about this, I strongly recommend that you contact a trademark attorney.
This answer is for general information purposes only and is not intended to be and should not be taken as legal advice. No attorney-client relationship or obligations are established herein, although consulting an attorney to discuss your specific situation is strongly recommended.