I believe this is an institution-dependent issue. Different departments and schools may have very different approaches as to what is formally allowed. In this day and age, however, collaborations, particularly between different departments or institutions, is generally looked on favorably. Whether it rises to an "official" arrangement or not, however, is clearly a different matter.
I don't believe that this is normally done via an employment contract; I think this is handled according to the regulations of the department—so the chair or graduate officer (or whoever normally approves student-advisor matches) would have to sign off on the arrangement. The responsibility for making sure all appropriate procedures are followed, however, lies with the advising team. (In general, this only matters with respect to the thesis committee and matters related to the thesis defense itself.)
The only other situation which might complicate matters is if both advisors are jointly funding the student; then the situation is certainly thornier. However, for such purposes, usually one advisor is the "hosting" advisor who has the primary responsibility for the student, which normally obviates the need for a complicated contract.