Co-teaching is required for public programs, and highly recommended for a 2-day client program or invite-only program. We recommend that corporate programs are co-taught (especially if the group size is larger than ~15 people), but this is not a requirement. For a half-day or keynote, teaching alone is just fine.
How much I should pay them?
When you hire a co-teacher, you are free to make whatever financial arrangement you'd like (and that works for them). This could be paying them a teaching fee, or sharing revenue, or creating a longer term partnership. SIYLI pays $1,250 USD per teaching day for most of our teachers ($750 for a half-day), $2,000 USD per day for our very experienced teachers, and $800 per travel day. The travel days are to compensate teachers for their travel time-- for example, a flight of more than 4 hours would be paid a full day, for less than 4 hours of travel time, SIYLI pays a half day rate.
You are welcome to use these rates, or offer something different. Please do adjust this to your local market rates! Alternatively, you might do a percentage revenue share with your co-teacher rather than a flat rate.
Should we create a contract?
I recommend that you create a contract between you and your client, and then a scope of work between you and your co-teacher. Here's a template that you can use (and feel free to modify) for hiring a coteacher.
Note: this article refers to paid client programs (not TT practicums or pro bono programs).