It is definitely the owner's responsibility, always the name of the person(s) on the deed and the town tax records. The bills ought to be sent to the address listed on the tax records to cover your butt. We have someone with an address that we know is different than the one listed in the tax records, so we send to both: the address we know they get it at, and the one listed on the tax records just to make sure our bases are covered.
Even if you do get payment from the renter per their agreement, you should still make sure the owner receives all proper documentation (meeting notices, minutes, paid invoices, etc) since they are ultimately the one responsible. If you end up taking someone to court, it will be the owner, not the renter, and if they haven't been receiving said documentation a judge could rule the debt null and void. It might be helpful to remind the owner in a notice of claim letter you send that that claim will have 0 impact on the renter, but could only hurt the credit rating of the owner.