Generally if your cost sharing arraingment is not with an intent to make a profit, you report nothing. You're just sharing expenses, and that's it. If there is a need for the person sharing expenses with you to show those expenses in the form of their housing costs, then since they don't own the home they will have to show they paid and/or are paying rent, and you will need to do exactly that - rent to them. You'll probably need a formal and legally recognized rental contract which isn't all that hard. On top of that, you have to report/claim the income on your tax return.
Then depending on what state you're in, you not only have to report that income on your state return. You may also have to conform to any landlord/tenant laws of your state, as well as any applicable ordinances of your particular locale within the state. Some states offer a "renter's credit" which means you have to register with the state or other locale within the state as a landlord/rental property owner. This usually requires you to issue state tax documents each year to the tenant so they can claim their renter's credit on their state tax return.
Then again, depending on the requirements of your state when it comes to state provided assistance, none of this may be necessary at all. You'll need to check with legal counsel regarding this, in your specific locale.
