There are typically two elements to a sale of chometz. The first and primary part is the sale of the actual chometz. The main focus of your typical mechiras chometz contract focuses on getting rid of the chometz, and not owning chometz on Pesach. This must be done by the owner of the chometz or someone acting through his/her agency. One cannot sell something that they do not own. (In fact, one of the thrusts of the contract is making the Rabbi your agent, since he cannot sell your chometz without being your agent.) So, unless you made arrangements with your co-owner as to the transference of authority to sell one another's chometz, the form would need to be filled out individually.
There is a second element of the sale, which is the lease of the locations in which the chometz is found, in order to not have chometz in your property on Pesach. If you are co-owners of the house, then this would depend on the terms of your partnership in ownership (setting aside the possibility of agency discussed above). If either, or both, of you have jurisdiction to make unilateral lease agreements, then the location is leased via one person's sale. If you both need to agree (which is far more typical), then you would both need to participate.
Note, as well, that the lease of property is, in most contracts, limited to the locations in which you have chometz. Therefore, unless you and your co-owner happen to only keep chometz in exactly overlapping locations, you would, regardless of the above, each need to separately lease the locations in which you have chometz as it probably does not coincide with the locations that your co-owner is leasing. (This could easily be changed by altering the contract, but generally these sales are pretty streamlined, especially when performed through online forms.)
In conclusion, the chometz is the primary focus of the sale, and probably simpler for each co-owner to do their own sale.