Most agencies do issue 1099s for stipends paid to host families for exchange students, if it's a flat monthly amount as opposed to reimbursement for specific expenses. You should verify with the agency that they will not be sending a 1099.
If you're being reimbursed for specific expenses that you incurred because of your guest, and you had to account to the agency involved in order to be paid, then there is no income to report.
Flat monthly amounts with no accountability to the agency are not specifically excluded from tax, and they would normally be reported as Non Employee Compensation. You can, however, deduct your actual expenses from the stipend, and if your expenses are equal to your stipend, you would not owe any tax.
If your expenses were greater than the stipend, you may be able to claim a charitable deduction for your extra hosting expenses. See page 4 of the following link:
https://www.irs.gov/pub/irs-pdf/p526.pdf
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