According to the IRS General Instructions for Certain Information Returns (Forms 1097, 1098, 1099, 3921, 3922, 5498, and W-2G), page 2:

A. Who Must File

See the separate specific instructions for each form.

Nominee/middleman returns. Generally, if you receive a Form 1099 for amounts that actually belong to another person, you are considered a nominee recipient. You must file a Form 1099 with the IRS (the same type of Form 1099 you received) for each of the other owners showing the amounts allocable to each. You also must furnish a Form 1099 to each of the other owners. File the new Form 1099 with Form 1096 with the Internal Revenue Service Center for your area. On each new Form 1099, list yourself as the “payer” and the other owner as the “recipient.” On Form 1096, list yourself as the “Filer.” A spouse is not required to file a nominee return to show amounts owned by the other spouse. The nominee, not the original payer, is responsible for filing the subsequent Forms 1099 to show the amount allocable to each owner.

To complete a Form 1099-MISC to report nominee income to the individual some (or all) of the income rightfully belongs to, you will use the IRS Instructions for Form 1099-MISC.

Form 1099-MISC Miscellaneous Income shows you the actual form (note that this form is for informational purposes only).  You cannot file this form but can obtain copies that can be completed and mailed to the IRS by contacting the IRS per the instructions on this form or visiting your local IRS office.

For instructions on sending the form to the correct agencies (and also sending the Form 1096 Annual Summary and Transmittal of U.S. Information Returns), you will return to the IRS General Instructions for Certain Information Returns (Forms 1097, 1098, 1099, 3921, 3922, 5498, and W-2G).  These instructions will also lead you to completing Form 1096 to send with the Form 1099-MISC.

Form 1096 Annual Summary and Transmittal of U.S. Information Returns (again, as explained above for Form 1099-MISC, this shows you the actual form but it is for informational purposes only). You cannot file this form but can obtain copies that can be completed and mailed to the IRS by contacting the IRS per the instructions on this form or visiting your local IRS office.

Sending Form 1099-MISC to the correct individual and sending Form 1096 and a copy of the Form 1099-MISC to the IRS provides them the ownership information so they can determine which return will contain the income.