With or without a written agreement, two or more people involved in a business for profit = a partnership. (Many states have a uniform partnership statute which creates a "one size fits all" agreement for these situations.) To maintain an account that way, you would need an EIN, a partnership resolution and any other documentation that your bank would require.
If Jack and Jill just have the checks made out to themselves from now on, this all goes away... As you have explained it, the relationship is actually that of two sole proprietors sharing expenses, not a partnership. However, they are confusing the issue by having checks made payable to a nonexistent business.
_________________________
In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.